Extrema

Occur at ______

• critical points where f’ changes from positive to negative (or vice versa)

• f(c) >/= f(x)

Maximum extrema

• f(c) </= f(x)

• Minimum extrema

First Derivative Test

• If c is a critical value of f → aka f’(c) = 0 or DNE

• f’(x) changes signs (positive/negative)

tests for max and min

First Derivative Test max

• F’(x) changes from positive to negative at x = c, f has a relative max at (c, f(c)) 

First Derivative Test min

• F’(x) changes from negative to positive at x = c, f has a relative min at (c, f(c)) 

Second Derivative Test

• Test for relative extrema 

• f’(c) = 0 and f”(c) exists

Second Derivative Test min

• f”(c) > 0 → min

Second Derivative Test max

f”(c) < 0 → max

Absolute extrema can only be

 critical points and endpoints

Candidates Test

tests for absolute min/max

Use x points to find y value

Extreme Value Theorem

• If f is continuous on a closed interval [a, b] 

• F has a minimum and maximum on the interval [a, b] 

• Always an absolute maximum and minimum

  • Critical points or endpoints