Question

A differentiable function f(x) will have a local minimum at x = b if -

• A. f’(b) = 0 , f’(b-h) < 0 & f’(b+h) > 0
• B. f’(b) = 0 , f’(b-h) > 0 & f’(b+h) < 0
• C. f’(b) = 0
• D. None of the above

Answer 1

The correct option is A

f’(b) = 0 , f’(b-h) < 0 & f’(b+h) > 0

We saw that if a function f(x) has a local maximum at x = c, then f’(c) = 0 , f’(c-h) > 0 & f’(c+h) < 0 . Similarly if x =b is a local minimum, then the derivative f’(b) will be zero. This is same as saying the slope of the tangent at x =b will be zero, which is clear from the figure. Now the sign of f’(x) on the left of b will be negative and on the right it will be positive.

So, we can say f’(b) = 0 , f’(b-h) < 0 & f’(b+h) > 0