Provide an appropriate response.If f(x) is a differentiable function and f' (c) = 0 at an interior point c of f's domain, and if for all x in the domain, must f have a local minimum at x = c? Explain.

Question

What will be an ideal response?

Answer 1

Yes. The point x = c is either a local maximum, a local minimum, or an inflection point. But, since   for all x in the domain, there are no inflection points and the curve is everywhere concave up and thus cannot have a local maximum. Hence, there is a local minimum at x = c.