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Calculus I









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Calculus I: Derivative
The Second Derivative Test





1. The Second Derivative Test

We say that x = c is a critical point of the function f(x) if f(c) exists and if either of the following are true.

f'(c) = 0 or f'(c) doesn't exist.

The Second Derivative Test tells us whether a critical point is a relative minimum or maximum.

The Second Derivative Test:

If x = c is a critical point at which f'(c) = 0 , 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 informative. That is x = c can be a relative maximum, relative minimum or neither.



2. Example

f(x)= (1/3)x3 - 3 x2 + 8 x + 5

so

f'(x) = x2 - 6 x + 8

The critical points are given by :

f'(x) = 0

Which are x1 = 2 and x2 = 4

f"(x) = 2 x - 6

• f"(2) = 2 (2) - 6 = 4 - 6 = - 2 < 0, hence

f has a relative maximum at x = 2 .

• f"(4) = 2 (4) - 6 = 8 - 6 = 2 > 0, hence

f has a relative minimum at x = 4 .








  


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