MC Questions Calculus

Question

(a) If f '(1) = 0 and f ''(1) = −3,

 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.

(b) If f '(3) = 0 and f ''(3) = 0,

 what can you say about f ?

1.At x = 3, f has a local maximum.

2.At x = 3, f has a local minimum.

   3. At x = 3, f has neither a maximum nor a minimum.

4.More information is needed to determine if f has a maximum or minimum at x = 3.

Answer 1

With all due respect to Doug C., I would like to add to his answer.

First,as Doug pointed out, when f'(a) = 0, the curve of f has an horizontal tangent.

The important point is that this may or may not be a maximum or minimum point.

What we know is that if f has a local maximum or a local minimum at a, then f'(a)=0.

The converse is not necessarily true...although in problems we often act as if it is!!!!!

Second, when both f'(a) AND f"(a) are 0, the curve of f still has a horizontal tangent at a, but this is the important case when an horizontal tangent is NOT at a local maximum or local minimum, but at a point of inflection with an horizontal tangent. Best example: f(x)=x³ at x=0.

Answer 2

For part a, the fact that f'(1) = 0 implies a horizontal tangent line at x = 1. Since we are only given the value of the 2nd derivative at x = 1, the 2nd derivative test must be used to determine if there is a local max or min there. Since f''(1) = -3, the 2nd derivative test states that there is a local max at x = 1. Think of it this way. Since f'' is the derivative of f' and f'' is negative, the slopes of tangent lines are decreasing at x = 1 (perhaps going from positive to negative? Or the curve lies under its tangent lines.

For b, since f'' = 0, the 2nd derivative test cannot be used. There is not enough information supplied to use the first derivative test. so more info is needed.