Exercise 7

Based on the given information about each function, decide whether the function has global maximum, a global minimum, neither, both, or that it is not possible to say without more information. Assume that each function is twice differentiable and defined for all real numbers, unless noted otherwise. In each case, write one sentence to explain your conclusion.

1. $f$ is a function such that $f''(x) \lt 0$ for every $x$.

2. $g$ is a function with two critical numbers $a$ and $b$ (where $a \lt b$), and $g'(x) \lt 0$ for $x \lt a$, $g'(x) \lt 0$ for $a \lt x \lt b$, and $g'(x) \gt 0$ for $x \gt b$.

3. $h$ is a function with two critical numbers $a$ and $b$ (where $a \lt b$), and $h'(x) \lt 0$ for $x \lt a$, $h'(x) \gt 0$ for $a \lt x \lt b$, and $h'(x) \lt 0$ for $x \gt b$. In addition, $\lim_{x \to \infty} h(x) = 0$ and $\lim_{x \to -\infty} h(x) = 0$.

4. $p$ is a function differentiable everywhere except at $x = a$ and $p''(x) \gt 0$ for $x \lt a$ and $p''(x) \lt 0$ for $x \gt a$.