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MS2 Max and Min Points

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MS2 Max and Min Points

  1. 1. Section 5.1 Increasing and Decreasing Functions
  2. 2. ObjectivesUpon completion of this lesson, you should be able to: • find where functions are increasing or decreasing
  3. 3. Increasing/Decreasing FunctionsSo far, we have only been able to determine if afunction is increasing or decreasing by plottingpoints to graph the function.Now that we know how to find the derivative of afunction, we will learn how the derivative can beused to determine the intervals where a function isincreasing or decreasing.
  4. 4. Increasing/Decreasing FunctionsRemember, the derivative of a function representsthe slope of the tangent line at a particular point onthe graph.So, if the derivative is positive on an open interval(a, b), then the slope of the tangent line is positive,which means the function is increasing on theinterval (a, b).So, if the derivative is negative on an open interval(a, b), then the slope of the tangent line is negative,which means the function is decreasing on theinterval (a, b).
  5. 5. Increasing/Decreasing FunctionsA function f is increasing on (a, b) if f (x1) < f (x2)whenever x1 < x2.A function f is decreasing on (a, b) if f (x1) > f (x2)whenever x1 < x2. Increasing Decreasing Increasing
  6. 6. Increasing/Decreasing/Constant Functions If f ′( x ) > 0 for each value of x in an interval ( a, b ) , then f is increasing on ( a, b ). If f ′( x ) < 0 for each value of x in an interval ( a, b ) , then f is decreasing on ( a, b ). If f ′( x ) = 0 for each value of x in an interval ( a, b ) , then f is constant on ( a, b ).
  7. 7. ExampleIn the given graph of the function f(x), determine theinterval(s) where the function is increasing,decreasing, or constant.
  8. 8. ExampleSolution:Looking at the graph from left to right, we wouldhave the following three intervals.The function is decreasing on the interval (-4, -2)The function is increasing on the interval (-2, 0)The function is decreasing on the interval (0, 2)
  9. 9. Critical NumbersIn order to find the intervals where a function isincreasing, decreasing, or constant without firstgraph the function, we must find what are calledcritical numbers.The critical numbers are those contained in thedomain of f(x) and which make the first derivativeequal to zero or undefined.
  10. 10. Critical Points of fA critical point of a function f is a point in thedomain of f where f ′( x) = 0 or f ′( x) does not exist. (horizontal tangent lines, vertical tangent lines and sharp corners)
  11. 11. Increasing/Decreasing FunctionsSteps in determining where a function is increasing ordecreasing:1. Find the derivative of the given function.2. Locate any critical numbers by seeing where the derivative is either zero or undefined.3. Plot the critical numbers on a number line to determine the open intervals.4. Select a test point in each interval and evaluate the derivative at this point.5. Use the sign of the derivative in each interval to determine whether it is increasing or decreasing.
  12. 12. ExampleDetermine the intervals where f ( x) = x 3 − 6 x 2 + 1is increasing and where it is decreasing. f ′( x) = 3x 2 − 12 x 3x 2 − 12 x = 0 3 x ( x − 4) = 0 3x = 0 or x − 4 = 0 x = 0, 4 + - + 0 4 f is decreasingf is increasing on ( 0, 4 )on ( −∞, 0 ) ∪ ( 4, ∞ )

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