Exercise 9

For each of the functions described below (each continuous on $[a,b]$), state the location of the function's absolute maximum and absolute minimum on the interval $[a,b]$, or say there is not enough information provided to make a conclusion. Assume that any critical numbers mentioned in the problem statement represent all of the critical numbers the function has in $[a,b]$. In each case, write one sentence to explain your answer.

1. $f'(x) \le 0$ for all $x$ in $[a,b]$

2. $g$ has a critical number at $c$ such that $a \lt c\lt b$ and $g'(x) \gt 0$ for $x \lt c$ and $g'(x) \lt 0$ for $x \gt c$

3. $h(a) = h(b)$ and $h''(x) \lt 0$ for all $x$ in $[a,b]$

4. $p(a) \gt 0$, $p(b) \lt 0$, and for the critical number $c$ such that $a \lt c \lt b$, $p'(x) \lt 0$ for $x \lt c$ and $p'(x) \gt 0$ for $x \gt c$