$\text { Use a graphing utility to graph } f ( x ) = \frac { 12 } { \sqrt [ 3 ] { x } } \text { and } P _ { 1 } \text {, a first-degree polynomial }$ function whose value and slope agree with the value and slope of f at .

Question

Quizplus · Accepted Answer

Using a graphing calculator, we can see that the graph of $f(x)$ is a curve that approaches infinity as $x$ approaches 0.

To find a polynomial function that matches the value and slope of $f(x)$ at $x=8,$ we need to find the value and slope of $f(x)$ at that point.

We have $f(x) = \frac{12}{\sqrt[3]{x}}.$ So at $x=8,$ we have

$f(8) = \frac{12}{\sqrt[3]{8}} = \frac{12}{2} = 6,$

and

$f'(x) = -4x^{-\frac{2}{3}}$

$f'(8) = -4\cdot (8)^{-\frac{2}{3}} = -1.$

So we want a line with slope $-1$ that passes through the point $(8,6).$

Using point-slope form, we get

$y-6 = -1(x-8),$

or

$y = -x+14.$

Thus, the function we want is $P_1(x) = -x+14,$ and the graph of $P_1(x)$ is the line passing through $(8,6)$ with slope $-1.$

Out of the answer choices, only graph D shows a line passing through $(8,6)$ with slope $-1.$