Question

If f(x) and f’(x) are differentiable at x = c, then the necessary condition for f(c) to be an extremum of f(x) is -

• A. f(c) = 0
• B. f’(c) = 0
• C. f”(c) = 0
• D. None of these

Answer 1

The correct option is B

f’(c) = 0

We know that if a function has an extremum at a point x = a and also the function is differentiable at x = a then the necessary condition would be f’(c) = 0. Notice that this is only necessary condition and not sufficient, meaning if f’(c) = 0 that doesn’t imply that “c” is an extrema. For example - $f\left(x\right)=x^3{f}^{\prime }\left(x\right)=3\times 2f^{\prime }\left(0\right)=0$ But we know that f(x) doesn’t have any extrema at x = 0.