Which of the following statements are true?

I. At a local maximum, $f'$ changes sign from positive to negative.

II. A function can have a local minimum value greater than one of its local maximum values.

III. A local maximum value is always the greatest value of the function.

IV. If $f$ has a local minimum at $c$ and $f'(c)$ exists, then $f'(c) = 0$.

Explanation

1. 01

   I is the first derivative test, and IV holds because the tangent is flat at a peak or trough of a differentiable function. For II: $x + \frac{1}{x}$ has a local maximum value $-2$ at $x = -1$ and a local minimum value 2 at $x = 1$.

2. 02

   III is false: 'local' means only among nearby points. $x^3 - 3x$ has a local maximum value 2 at $x = -1$, yet $f(3) = 18$.