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 maximum 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 B

f”(c) < 0

We saw that the slope of the tangents just before a local maximum is positive and just after local maximum, it is negative. Also, at maximum or 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) decreases. That is, before x =b, it is positive, at x=b, it is zero and after x =b, it is negative. Since the value of f’(x) decreases, we can say that the rate of change of f’(x), f”(x), will be negative at x =c

Or f”(c) <0

=> option b is correct