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MAXIMUM AND MINIMUM VALUES

MAXIMUM AND MINIMUM VALUES. A critical number is the value of x that makes f ′ ( x ) = 0 or f ′ ( x ) undefined The absolute minimum of a function is the very lowest point on the graph over a given interval of x .

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MAXIMUM AND MINIMUM VALUES

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  1. MAXIMUM AND MINIMUM VALUES • A criticalnumber is the value of x that makes f ′(x) = 0 or f ′(x)undefined • The absoluteminimum of a function is the very lowest point on the graph over a given interval of x. • The absolutemaximum of a function is the very highest point on the graph over a given interval of x. • Localminimums and maximums are turning points on the graph. They are at the values of x for which f ′(x)= 0.

  2. This function is graphed over the interval -2 <x< 7 The absolute maximum value is 5 The local maximum valuesare 2 and 3 The local minimum values are -2 and 1 The absolute minimum value is -2

  3. Procedure for Finding the Absolute Maximum and Minimum Values of a Continuous Function on a Closed Interval [a, b] • Find the values of f (x)where f ' (x) = 0 • Find the values of f (a)and f (b). • The largest of the values from above is the absolute maximum value 4. The smallest of these values if the absolute minimum value

  4. EXAMPLE 1 Graph the function f(x) = x2 – 2x – 8 over the domain –2<x<3 State the absolute and local maximum and minimum values f ' (x) = 2x – 2 2x – 2 = 0 when x = 1 Turning point is at f(1) = -9 absolute maximum is 0 local and absolute minimum is - 9

  5. EXAMPLE 2 Graph the function f(x) = x3 – 9xover the domain –4<x<3 State the absolute and local maximum and minimum values Find the critical values where the slope of the tangent is 0. f ' (x) = 3x2 – 9 x2 = 3 3x2 – 9 = 0 3x2 = 9

  6. EXAMPLE 2 Graph the function f(x) = x3 – 9xover the domain –4<x<3 State the absolute and local maximum and minimum values local and absolute max is 10.4 local minimum is -10.4 absolute minimum is -28

  7. Homework Assignment Lesson #2 Maximum and Minimum Values Complete Questions 1 – 2 on Pages 4 – 6 Check all your work carefully with your calculator graphs.

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