Question

f(x) and f’(x) are differentiable at x = c. A sufficient condition for f(c) to be an extremum of f(x) is that f’(x) changes sign as x passes through c

• A. True
• B. False

Answer 1

The correct option is A

True

We know that f(x) will always have an extremum if f’(x) changes its sign in the neighborhood of a point. If f’(c+h) < 0 & f’(c-h) > 0 then we will have a maximum at x = c given that the function is differentiable at x =c. And if f’(c+h) > 0 & f’(c-h) < 0 then we will have a minimum at x = c.

So, if we know that f’(x) is changing its sign in the neighborhood of some point “c” and also f(x) is differentiable at x = c then we can say that f(x) has an extremum at x = c.

[h is tending to zero from right side]