Lecture 12
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This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.
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Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
3.1 extrema on an interval
The document defines extrema as the extreme or maximum and minimum values of a function on an interval. An absolute extremum occurs at an endpoint of a closed interval, while a relative extremum occurs in the interior of an open interval. A function has extrema at points where the derivative is equal to 0 (critical points) or where the function is not differentiable. The Extreme Value Theorem states that a continuous function on a closed interval will have both an absolute maximum and minimum. To find the extrema of such a function, evaluate it at all critical points and endpoints. A function can have a maximum or minimum at multiple points within an interval.
Critical points
The document discusses local maxima and minima of functions. It defines a local maximum/minimum as a point where the function values in an interval around that point are less than/greater than the function value at that point. Critical points, where the derivative is zero, are important for determining local extrema. Two tests are described - the first derivative test examines the sign of the derivative on each side of the critical point, and the second derivative test uses the concavity at the critical point to determine if it is a maximum or minimum. Examples are provided to illustrate applying these tests.
Extreme values of a function & applications of derivative
This document discusses key concepts related to finding extrema of functions, including:
- Absolute and relative extrema refer to the maximum and minimum values of a function over its entire domain or on a subinterval, respectively.
- Critical points, where the derivative is zero or undefined, and endpoints must be checked to find extrema.
- The Extreme Value Theorem states that continuous functions on closed intervals have both a maximum and minimum value.
- The first derivative determines whether a function is increasing or decreasing, and where it is zero may indicate relative extrema. The second derivative indicates concavity and points of inflection.
5.2 first and second derivative test
This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
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This document summarizes a lesson on critical numbers:
1) Critical numbers indicate local maximums and minimums, and are found by setting the derivative f'(x) equal to 0.
2) A critical number is a number C where either f'(C) = 0 or f'(C) does not exist.
3) Not all critical points are local extrema - some are points of inflection where the second derivative changes sign.
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Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
3.1 extrema on an interval
The document defines extrema as the extreme or maximum and minimum values of a function on an interval. An absolute extremum occurs at an endpoint of a closed interval, while a relative extremum occurs in the interior of an open interval. A function has extrema at points where the derivative is equal to 0 (critical points) or where the function is not differentiable. The Extreme Value Theorem states that a continuous function on a closed interval will have both an absolute maximum and minimum. To find the extrema of such a function, evaluate it at all critical points and endpoints. A function can have a maximum or minimum at multiple points within an interval.
Critical points
The document discusses local maxima and minima of functions. It defines a local maximum/minimum as a point where the function values in an interval around that point are less than/greater than the function value at that point. Critical points, where the derivative is zero, are important for determining local extrema. Two tests are described - the first derivative test examines the sign of the derivative on each side of the critical point, and the second derivative test uses the concavity at the critical point to determine if it is a maximum or minimum. Examples are provided to illustrate applying these tests.
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- Absolute and relative extrema refer to the maximum and minimum values of a function over its entire domain or on a subinterval, respectively.
- Critical points, where the derivative is zero or undefined, and endpoints must be checked to find extrema.
- The Extreme Value Theorem states that continuous functions on closed intervals have both a maximum and minimum value.
- The first derivative determines whether a function is increasing or decreasing, and where it is zero may indicate relative extrema. The second derivative indicates concavity and points of inflection.
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This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
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This document summarizes a lesson on critical numbers:
1) Critical numbers indicate local maximums and minimums, and are found by setting the derivative f'(x) equal to 0.
2) A critical number is a number C where either f'(C) = 0 or f'(C) does not exist.
3) Not all critical points are local extrema - some are points of inflection where the second derivative changes sign.
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Lesson 4.3 First and Second Derivative Theory
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2) A curve is concave up if the second derivative f''(x) is positive, and concave down if f''(x) is negative. To determine concavity, examine the sign of f''(x) around any points where it is zero or undefined. A change in sign indicates an inflection point where the concavity changes.
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
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3) Finally, the document outlines the test for concavity and points of inflection based on the sign of the second derivative on an interval. A change in the sign of the second derivative indicates an inflection point.
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Second Derivative Information
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1. The document defines extrema as the minimum and maximum values of a function over an interval. On a closed interval, a continuous function is guaranteed by the Extreme Value Theorem to have both a minimum and maximum value.
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This document discusses key concepts related to finding extrema of functions:
- Absolute and relative extrema refer to the maximum and minimum values of a function over its entire domain or on a sub-interval, respectively. Critical points where the derivative is zero or undefined must be checked.
- The Extreme Value Theorem states that if a function is continuous on a closed interval, it will have both a maximum and minimum value on that interval.
- To find extrema, one finds all critical points, checks the endpoints, and compares the function values to determine the maximum and minimum. The derivative indicates whether a function is increasing or decreasing and points where it is zero or undefined.
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1) To determine if a function is increasing or decreasing, examine the sign of the first derivative f'(x) at points around any critical points. A change from positive to negative indicates a relative maximum, and a change from negative to positive indicates a relative minimum.
2) A curve is concave up if the second derivative f''(x) is positive, and concave down if f''(x) is negative. To determine concavity, examine the sign of f''(x) around any points where it is zero or undefined. A change in sign indicates an inflection point where the concavity changes.
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This document provides an overview of concavity, inflection points, and how to find the local extrema of functions. It explains that a graph is concave up when tangent slopes are increasing and concave down when slopes are decreasing. An inflection point is where the direction of concavity changes and occurs when the second derivative is zero or undefined, though one must check the signs of the second derivative on either side. It gives examples of finding the inflection points of f(x)=x^4+2x^3-1 and using the second derivative test to find local extrema of f(x)=x^2e^x, and provides practice problems from the textbook.
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The second derivative of a function f(x) provides information about the concavity and points of inflection of f(x). It indicates the intervals where f(x) is concave up or down. A curve is concave up where the second derivative is positive and concave down where the second derivative is negative. Points of inflection are where the concavity changes from up to down or vice versa, and occur where the second derivative is zero or undefined.
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3) The second derivative determines concavity and points where it changes sign indicate inflection points.
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We define what it means for a function to have a maximum or minimum value, and explain the Extreme Value Theorem, which indicates these maxima and minima must be there under certain conditions.
Fermat's Theorem says that at differentiable extreme points, the derivative should be zero, and thus we arrive at a technique for finding extrema: look among the endpoints of the domain of definition and the critical points of the function.
There's also a little digression on Fermat's Last theorem, which is not related to calculus but is a big deal in recent mathematical history.
Lesson 18: Maximum and Minimum Vaues
This document discusses finding the maximum and minimum values of functions. It introduces the Extreme Value Theorem, which states that if a function is continuous on a closed interval, then it attains both a maximum and minimum value on that interval. It also discusses Fermat's Theorem, which relates local extrema of a differentiable function to its derivative. Examples are provided to illustrate these concepts.
Derivatives in graphing-dfs
This document provides an overview of key concepts in calculus related to derivatives, including: analyzing functions to determine if they are increasing or decreasing; finding relative extrema, critical points, and inflection points; using the first and second derivative tests to determine concavity; and graphing polynomials. Examples are provided to illustrate how to apply these concepts to specific functions in order to analyze intervals of increase/decrease, locate critical points, identify relative maxima and minima, and determine intervals of concavity. Videos and Khan Academy links are also included for supplemental instruction on related topics.
Lesson 18: Maximum and Minimum Values (Section 021 handout)
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson19 Maximum And Minimum Values 034 Slides
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
maxima & Minima thoeyr&solved.Module-4pdf
In this chapter we shall study those points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards. At such points the derivative of the function, if it exists, is necessarily zero.
The value of a function f (x) is said to be maximum at x = a, if there exists a very small positive number h, such that
f(x) < f(a) x (a – h,a + h) , x a
In this case the point x = a is called a point of maxima for the function f(x).
Simlarly, the value of f(x) is said to the minimum at x = b, If there exists a very small positive number, h, such that
f(x) > f(b), x (b – h,b + h), x b
In this case x = b is called the point of minima for the function f(x).
Hene we find that,
(i) x = a is a maximum point of f(x)
f(a) f(a h) 0
f(a) f(a h) 0
(ii) x = b is a minimum point of f(x)
Rf(b) f(b h) 0
f(b) f(b h) 0
(iii) x = c is neither a maximum point nor a minimum point
Note :
(i) The maximum and minimum points are also known as extreme points.
(ii) A function may have more than one maximum and minimum points.
(iii) A maximum value of a function f(x) in an interval [a,b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function. A minimum value may be greater than some maximum value for a function.
(iv) If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
(v) Monotonic functions do not have extreme points.
Ex. Function y = sin x, x (0, ) has a maximum point at x = /2 because the value of sin /2 is greatest in the given interval for sin x.
Clearly function y = sin x is increasing in the interval (0, /2) and decreasing in the interval ( /2, ) for that reason also it has maxima at x = /2. Similarly we can see from the graph of cos x which has a minimum point at x = .
Ex. f(x) = x2 , x (–1,1) has a minimum point at x = 0 because at x = 0, the value of x2 is 0, which is
less than the all the values of function at different points of the interval.
Clearly function y = x2 is decreasing in the interval
Rf(c) f(c h)
Sand
Tf(c) f(c h) 0
|V| have opposite signs.
(–1, 0) and increasing in the interval (0,1) So it has minima at x = 0.
Ex. f(x) = |x| has a minimum point at x = 0. It can be easily observed from its graph.
A. Necessary Condition : A point x = a is an extreme point of a function f(x) if f’(a) = 0, provided f’(a) exists. Thus if f’ (a) exists, then
x = a is an extreme point f’(a) = 0
or
f’ (a) 0 x = a is not an extreme point.
But its converse is not true i.e.
f’ (a) = 0 x = a is an extreme point.
For example if f(x) = x3 , then f’ (0) = 0 but x = 0 is not an extreme point.
B. Sufficient Condition :
(i) The value of the function f(x) at x = a is maximum, if f’ (a) = 0 and f” (a) < 0.
(ii) The value of
Lesson 19: Maximum and Minimum Values
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- 2. Def: Absolute Extreme Values Let f be a function with domain D. Then f(c) is the… a) Absolute (or global) maximum value on D iff f(x) ≤ f(c) for all x in D. b) Absolute (or global) minimum value on D iff f(x) ≥ f(c) for all x in D. * Note: The max or min VALUE refers to the y value.
- 3. Where can absolute extrema occur? * Extrema occur at endpoints or interior points in an interval. * A function does not have to have a max or min on an interval. * Continuity affects the existence of extrema. Min Max Min No Max No Max No Min
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- 5. Def: Local Extreme Values Let c be an interior point of the domain of f. Then f(c) is a… a) Local (or relative) maximum value at c iff f(x) ≤ f(c) for all x in some open interval containing c. b) Local (or relative) minimum value at c iff f(x) ≥ f(c) for all x in some open interval containing c. * A local extreme value is a max or min in a “neighborhood” of points surrounding c
- 6. Identify Absolute and Local Extrema Abs & Loc Min Abs & Loc Max Loc Max Loc Min Loc Min * Local extremes are where f ’(x)=0 or f ’(x) DNE
- 7. Thm 2: Local Extreme Values If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f ’ exists at c, then f ’(c)=0.
- 8. Def: Critical Point A critical point is a point in the interior of the domain of f at which either… 1) f ’(x)= 0, or 2) f ’(x) does not exist
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