Question

Suppose f '' is continuous on (−∞, ∞). (a) if f '(1) = 0 and f ''(1) = −7, what can you say about f ? at x = 1, f has a local maximum. at x = 1, f has a local minimum. at x = 1, f has neither a maximum nor a minimum. more information is needed to determine if f has a maximum or minimum at x = 1.

Answer 1

If f'(1) = 0, then there must be a turning point at x = 1. We can deduce that by observing the fact that a particular point (x = 1), the tangent of f(x) will be horizontal. This implies that at some points after and before x = 1, there will be a change in curvature.

Now, if the second derivative < 0, then there is a maximum turning point. This can be deduced by the fact that negative second derivatives represent a point that has a negative concavity.

Thus, there will be a maximum point at f(1).
ie. at x = 1, f has a local maximum.