Q. 2 TB - Calculus

Question

Find the x-coordinates of the points where the graphs of $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-9{\mathrm{x}}^2$and $\mathrm{g}\left(\mathrm{x}\right)=\mathrm{x}-9$intersect.

Step-by-Step Solution

Answer

x-coordinates of the intersection points of $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-9{\mathrm{x}}^2$and $\mathrm{g}\left(\mathrm{x}\right)=\mathrm{x}-9$are $x=1,-1, 9$.

Step 1. Given information

Given functions are the followings.

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-9{\mathrm{x}}^2 \\ \mathrm{g}\left(\mathrm{x}\right)=\mathrm{x}-9 \end{gathered}$$

Step 2. Intersection points.

To determine the intersection points, equate both functions and solve the equation for x.

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right): \\ {\mathrm{x}}^3-9{\mathrm{x}}^2=\mathrm{x}-9 \\ {\mathrm{x}}^3-9{\mathrm{x}}^2-\mathrm{x}+9=0 \\ {\mathrm{x}}^2(\mathrm{x}-9)-1(\mathrm{x}-9)=0 \\ (\mathrm{x}-9)({\mathrm{x}}^2-1)=0 \\ \Rightarrow \mathrm{x}=9,\pm 1 \end{gathered}$$

So x-coordinates of the points where the graphs of functions $f\left(x\right)$ and $g\left(x\right)$intersect are $\mathrm{x}=9,1,-1$