Transcript
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.
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