### Home > APCALC > Chapter 4 > Lesson 4.1.3 > Problem4-37

4-37.

Sketch a graph of $f\left(x\right) = x^{3} – 2x^{2}$. At what point(s) will the line tangent to $f$ be parallel to the secant line through $\left(0,f\left(0\right)\right)$ and $\left(2,f\left(2\right)\right)$?

Calculate the slope of the secant between $\left(0, f\left(0\right)\right) \text{ and } \left(2, f\left(2\right)\right)$.

$\text{slope of secant }=\frac{f(2)-f(0)}{2-0}=\frac{0-0}{2}=0$

We want to know where the slope of the tangent is the same as the slope of the secant. Recall that the slope of the tangent is also
known as $f^\prime\left(x\right)$, find where $f^\prime\left(x\right) = 0$.

The slope of the tangent $=$ the slope of the secant at coordinate points ( ________, _________ ) and ( ________, _________ ). You must analytically compute the exact coordinates, but note that the slope of tangents lines is $0$ at the local maximum and local minimum.

Use the eTool below to examine the graph.
Click the link at right for the full version of the eTool: Calc 4-37 HW eTool