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
• B. f”(c) < 0
• C. f”(c) = 0
• D. None of the above

Answer 1

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