CPM Homework Help : APCALC Problem 6-36

Sketch a continuous function that satisfies all of the following conditions. Read carefully—some are limits of the derivative, not the function. Where is $f$ non-differentiable?

$\lim\limits _ { x \rightarrow - \infty } f ( x ) = \infty$

$\lim\limits _ { x \rightarrow 0 ^ { - } } f ^ { \prime } ( x ) = - 1$

$\lim\limits _ { x \rightarrow 0 ^ { + } } f ^ { \prime } ( x ) = 1$

$\lim\limits _ { x \rightarrow \infty } f ( x ) = 5$

Hint:

Clues 1 and 4 refer to the end behavior of $f\left(x\right)$ . Notice that there $f\left(x\right)$ approaches $∞$ in one direction, but approaches a horizontal asymptote in the other.

Hint:

Clues 2 and 3 refer the slopes before and after $x = 0$ . Notice that these limits describes the derivative, $f '\left(x\right),$ not the actual function that you are graphing. Also notice that the limit from the left does not agree with the limit from the right, so $f\left(x\right)$ is NOT differentiable at $x = 1$ .