    Question

# f(x) and f’(x) are differentiable at x = c. Which of the following is the condition for f(x) to have a local minimum at x = c, if f’(c) = 0

A

f”(c) > 0

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B

f”(c) < 0

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C

f”(c) = 0

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D

None of the above

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Solution

## The correct option is A f”(c) > 0 We saw that the slope of the tangents just before a local minimum is negative and just after local minimum, it is positive. Also, at maximum and minimum f’(x) will be zero. So, we have f’(c-h) <0 & f’(c+h) >0 Now, we know that f”(x) is the rate of change of f’(x). Around x =c, the value of f’(x) increases. That is, before x = c, it is negative, at x = c, it is zero and after x = c, it is positive. Since the value of f’(x) increases, we can say that the rate of change of f’(x), f”(x), will be positive at x = c Or f”(c) >0 => option a is correct  Suggest Corrections  0      Similar questions
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