5.2 first and second derivative test

Relative ExtremaFirst Derivative Test - FDTSecond Derivative Test - SDT

Relative Extrema

decreasing

decreasingincreasing

relative maximum

relative maximumrelative

minimumrelative minimum

Critical Points

Critical points are points at which:

•Derivative equals zero (also called stationary point).

•Derivative doesn’t exist.

First Derivative Test

Let f be a differentiable function with f '(c) = 0, then:

•If f '(x) changes from positive to negative, then f has a relative maximum at c.•If f '(x) changes from negative to positive, then f has a relative minimum at c.

•If f '(x) has the same sign from left to right, then f

does not have a relative extremum at c.

Practice Time!!!

Use First Derivative Test to find critical points and state whether they are minimums or maximums.

Critical points

0 2(stationary)

0 2

+ + __

relativemaximum

relativeminimum

Second Derivative Test

Suppose that c is a critical point at which f’(c) = 0, that f(x) exists in a neighborhood of c, and that f(c) exists. Then:

• f has a relative maximum value at c if f”(c) < 0.

•f has a relative minimum value at c if f”(c) > 0.

•If f(c) = 0, the test is not conclusive.

Note: Second derivative test is still used to calculate max and min

Practice Time again !!!Use second derivative test to find extrema

of

critical points = 0, -1, 1

inconclusive

f has a maximum at x = -1

f has a minimum at x = 1

Find extrema of

A.Using first derivative test

B.Using second derivative test