함수의 증가와 감소

For the following exercises, consider a third-degree polynomial $f(x),$ which has the properties ${f}^{'}(1)=0,{f}^{'}(3)=0$ (not applicable to Exercises 249 and 250). Determine whether the following statements are true or false. Justify your answer.

$f(x)=0$ for some $1\le x\le 3$

For the following exercises, consider a third-degree polynomial $f(x),$ which has the properties ${f}^{'}(1)=0,{f}^{'}(3)=0$ (not applicable to Exercises 249 and 250). Determine whether the following statements are true or false. Justify your answer.

${f}^{''}(x)=0$ for some $1\le x\le 3$

For the following exercises, consider a third-degree polynomial $f(x),$ which has the properties ${f}^{'}(1)=0,{f}^{'}(3)=0$ (not applicable to Exercises 249 and 250). Determine whether the following statements are true or false. Justify your answer.

There is no absolute maximum at $x=3$

In your own words describe what it means for a function to be increasing.

What does a decreasing function “look like”?

Sketch a graph of a function on $[0,2]$ that is increasing, where it is increasing “quickly” near $x=0$ and increasing “slowly” near $x=2$.

Give an example of a function describing a situation where it is “bad” to be increasing and “good” to be decreasing.

T/F: Functions always switch from increasing to decreasing, or decreasing to increasing, at critical points.

A function $f$ has derivative $f\,'(x) = (\sin x+2)e^{x^2+1}$, where $f\,'(x) >1$ for all $x$. Is $f$ increasing, decreasing, or can we not tell from the given information?