The document discusses extreme values (maximums and minimums) of functions. It defines local and absolute extrema and notes that not all functions have extrema. Critical points, where the derivative is zero or undefined, are discussed. The relationship between critical points and extrema is explained by Fermat's Theorem. Methods for finding extrema on closed intervals using critical points and endpoints are presented, along with an example. Rolle's Theorem relating critical points to functions with equal values at endpoints is also introduced.
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.
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.
Homomorphic Lower Digit Removal and Improved FHE Bootstrapping by Kyoohyung Hanvpnmentor
Kyoohyung Han is a PhD student in the Department of Mathematical Science at the Seoul National University in Korea. These are the slides from his presentation at EuroCrypt 2018.
This presentation is a part of Computer Oriented Numerical Method . Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.
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
Homomorphic Lower Digit Removal and Improved FHE Bootstrapping by Kyoohyung Hanvpnmentor
Kyoohyung Han is a PhD student in the Department of Mathematical Science at the Seoul National University in Korea. These are the slides from his presentation at EuroCrypt 2018.
This presentation is a part of Computer Oriented Numerical Method . Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.
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
A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
2. Applications of the Derivative
• One of the most common applications of the
derivative is to find maximum and/or minimum
values of a function
• These are called “Extreme Values” or “Extrema”
• Extrema would be an excellent name for an 80’s
Hair Band.
3. Definition
• Let f (x) be a function defined on an interval
I, let a ϵ I, then f (a) is
• Absolute minimum of f (x) on I, if
f (a) ≤ f (x) for all x in I
• Absolute maximum of f (x) on I, if
f (a) > f (x) for all x in I
• If no interval is indicated, then the extreme
values apply to the entire function over its
domain.
4. Do All Functions Have Extrema?
• f (x) = x
• No extrema unless the function is defined on an
interval.
5. Do All Functions Have Extrema?
• g (x) = (-x(x2 – 4))/x
• Discontinuous and has no max on [a, b]
a b
6. Do All Functions Have Extrema?
• f (x) = tan x
• No max or min on the open interval (a, b)
a b
7. Do All Functions Have Extrema?
• h(x) = 3x3 + 6x2 + x + 3
• Function is continuous and [a, b] is closed.
Function h(x) has a min and max.
b
a
8. Definition
Local Extrema a function f (x) has a:
• Local Minimum at x = c if f (c) is the
minimum value of f on some open interval (in
the domain of f) containing c.
• Local Maximum at x = c if f (c) is the
maximum value of f on some open interval
containing c.
10. Absolute and Local Max
(a, f(a)) (c, f (c))
Absolute max on [a,b] (b, f(b))
Local Max
a c b
11. Critical Points
Definition of Critical Points
• A number c in the domain of f is called a critical
point if either f’ (c) = 0 or f’ (c) is undefined.
12. Fermat’s Theorem
• Theorem: If f (c) is a local min or
max, then c is a critical point of f.
• Not all critical points yield local extrema. “False
positives” can occur meaning that f’(c) = 0 but
f(c) is not a local extremum.
13. Fermat’s Theorem
f(x) = x3 + 4
Tangent line at (0, 4) is horizontal
f(0) is NOT an extremum
14. Optimizing on a Closed Interval
Theorem: Extreme Values on a Closed
Interval
• Assume f (x) is continuous on [a, b] and let f(c)
be the minimum or maximum value on [a, b].
Then c is either a critical point or one of the
endpoints a or b.
15. Example
Find the extrema of f(x) = 2x3 – 15x2 + 24x + 7 on
[0, 6].
• Step 1: Set f’(x) = 0 to find critical points
▫ f’(x) = 6x2 – 30x + 24 = 0, x = 1, 4
• Step 2: Calculate f(x) at critical points and
endpoints.
▫ f(1) = 18, f(4) = -9, f(0) = 7, f(6) = 43
• The maximum of f(x) on [0, 6] is (6, 43) and
minimum is (4, -9).
16. Graph of f(x)=2x3-15x2+24x+7
Endpoint
Max (6, 43)
Critical Point – local
max (1, 18)
Endpoint
(0, 7)
Critical point – local min
(4, -9)
20. Example
• Critical points: g’(x) = cos 2 x – sin 2 x
• g’(x) = 0, x = π/4, 3π/4
• g(π/4) = ½ , max
• g(3π/4) = -1/2 , min
• Endpoints (0, 0), (π, 0)
21. Rolle’s Theorem
• Assume f (x) is continuous on [a, b] and
differentiable on (a, b). If f (a) = f (b) then there
exists a number c between a and b such that
f’(c) = 0
f(c)
f(a) f(b)
a c b
22. Example
• Use Rolle’s Theorem to show that the function
f(x) = x3 + 9x – 4 has at most 1 real root.
23. Example
• If f (x) had 2 real roots a and b, then f (a) = f (b)
and Rolle’s Theorem would apply with a number
c between a and b such that f’(c) = 0.
• However…f’(x) = 3x2 + 9 and 3x2 + 9 = 0 has no
real solutions, so there cannot be a value c such
that f’ (c) = 0 so there is not more than 1 real
root of f (x).