Chapter 4.5, Problem 61E

To determine

To solve: the equation $f\left(x\right)=g\left(x\right)$ for the given function graphically and algebraically.

Answer to Problem 61E

  $x=2$

Explanation of Solution

Given information:

Given functions are,

  $f\left(x\right)=x^3-4x^2+4x$

And

  $g\left(x\right)=-2x+4$

Calculation:

Consider the given functions.

  $f\left(x\right)=x^3-4x^2+4x$ …… (1)

  $g\left(x\right)=-2x+4$ …… (2)

Now, the points of intersection of the graph of $f\left(x\right)$ and $g\left(x\right)$ will be the solution of $f\left(x\right)=g\left(x\right)$ .

Construct the graphs of $f\left(x\right)$ and $g\left(x\right)$ in same set of axes.

  

Here, the point of intersection is $\left(2,0\right)$ .

Thus, the solution is $x=2$ .

Now, solve $f\left(x\right)=g\left(x\right)$ algebraically.

  $\begin{array}{c}{x}^3-4x^2+4x=-2x+4\\ x^3-4x^2+4x+2x-4=0\\ x^3-4x^2+6x-4=0\\ x^3-2x^2-2x^2+4x+2x-4=0\\ x^2\left(x-2\right)-2x\left(x-2\right)+2\left(x-2\right)=0\\ \left(x-2\right)\left(x^2-2x+2\right)=0\end{array}$

Now, $x^2-2x+2\ne 0$ for any real x .

So, $x-2=0$

Hence, $x=2$ is the only real solution of $f\left(x\right)=g\left(x\right)$ .