Question

In Problems 75, determine where the graph of $f$is below the graph of $g$by solving the inequality $f\left(x\right)\le g\left(x\right)$. Graph $f$and $g$together.

$$\begin{gathered} f\left(x\right)=x^4-4 \\ g\left(x\right)=3x^2 \end{gathered}$$

Short answer

The graph is shown below:

Step-by-step solution

Given information

The given function are shown below:

$$\begin{gathered} f\left(x\right)=x^4-4 \\ g\left(x\right)=3x^2 \end{gathered}$$

Determine the intersection point. 

For intersection point, $f\left(x\right)=g\left(x\right)$.

$\begin{array}{rcl}{x}^4-4& =& 3x^2\\ x^4-3x^2-4& =& 0\\ x^4-4x^2+x^2-4& =& 0\\ x^2\left(x^2-4\right)+1\left(x^2-4\right)& =& 0\\ \left(x^2+1\right)\left(x^2-4\right)& =& 0\\ \left(x^2+1\right)\left(x-2\right)\left(x+2\right)& =& 0\\ x& =& 2,-2\end{array}$

$x^2+1$ given the imaginary solution.

Graph the function fx

Sketch the graph of $y=f\left(x\right)$having vertex at $(0,-4)$and range $[-4,\infty )$.

Graph the function gx

Sketch the graph of $y=g\left(x\right)$having vertex at $\left(0,0\right)$and range $[0,\infty )$.

Determine the shaded portion which satisfy the given inequality.

The portion where $f\left(x\right)=x^4-4$less than or equal to $g\left(x\right)=3x^2$is in the interval of $[-2,2]$as shaded in the following figure.

Write the conclusion

The final graph is shown below: