Extrema on an Interval

Increasing and Decreasing Let f be a function that is continuous and differentiable.

1. If f’(x) > 0 , then f(x) is increasing2. If f’(x) < 0 , then f(x) is decreasing3. If f’(x) = 0 , then f(x) is constant

Critical Number Let f be defined at c. If f’(c) = 0 or if f’(c) is not differentiable at c, then c is a critical number of f.

Relative Extrema occurs ONLY at critical numbers If f has a relative minimum or relative maximum at x = c, then c is a critical number of f.

First Derivative TestLet c be a critical number of a function f that is continuous, then f(c) can be classified as follows.

1. If f’(x) changes from negative to positive at c, then f has a relative minimum at (c, f(c)).

2. If f’(x) changes from positive to negative at c, then f has a relative maximum at (c, f(c)).

3. If f’(x) does not change from negative to positive or positive to negative at c, then f(c) is neither a relative maximum or relative minimum.

Finding Relative ExtremaSteps:

1. Find the derivative.2. Find the critical numbers of f by finding when

f’(x)=0 or f’(x) is undefined.3. Test values in between the critical numbers.4. Classify critical points as a relative maximum,

relative minimum or neither.

Example 1Given the function above: Identify the critical numbers, find when the function is increasing and/or decreasing, and classify the critical numbers as a relative max, min or neither.

fYx7 6x2t6x fix has a rel men

f 4 7 6 1 1 X 0 b c f'Cx Changes

0 6 4 11 from to 1

6 0 Xtro f x has a rel Max

11 0 x I 4 1 b c fill changes

critical S from to

fixf X is increasing to

1 U

g O b c flex O

l f X is decreasing onC 1,0 BIC flex O

Example 2Given the function above: Identify the critical numbers, find when the function is increasing and/or decreasing, and classify the critical numbers as a relative max, min or neither.

F'Cx 2CxtXxt3 x D2 f x nasarelminx 1b1cfkX7

f x x 1 2 16 1 1 Changes from to 1

fCx has a rel Maxf 4 7 11 1 3 5

x l blotch0 1 1713 5 changes from 110

f I X 53 fCx is increasingis Es U l o ble

fYx fl x Oc to 0 1

1,17 fad is decreasingE 5 3,1 b c f 4 7 03

Absolute Extrema● Absolute Maximum - is the highest finite y-value

of a function.● Absolute Minimum - is the lowest finite y- value

of a function.

The Extreme Value TheoremIf f is continuous on a closed interval [a, b], then f has a both an absolute minimum and an absolute maximum on the interval.

To find Absolute Extrema Steps:

1. Find the derivative.2. Find the critical numbers of f by finding when

f’(x)=0 or f’(x) is undefined.3. Evaluate critical numbers and endpoints using f(x).4. The greatest value is the absolute maximum and

the absolute minimum is the smallest value.

Example 1 Find the absolute extrema of the function on the closed interval.

f 4 7 4 8 f O D0 4 8 f 2 8X 2 f 161 24

flex has an absolute maximum

X 6

ftx has an absolute minimum

11 2