Calculus I
Limits
Derivative
Exercices
Applications
Marginal analysis
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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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