91Ó°ÊÓ

The given functions are$f\left(x\right)=x^4-1$and$g\left(x\right)=-2x^2+2$.

Substitute $f\left(x\right)=x^4-1$and $g\left(x\right)=-2x^2+2$into the inequality $f\left(x\right)≤g\left(x\right)$.

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

Now, find the zeros of the function.

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

$x^2+3$gives the imaginary solution.

$(-∞,-1),(-1,1),(1,+∞)$

Select a test number in each interval found in above step and evaluate $\left(x-1\right)\left(x+1\right)\left(x^2+3\right)≤0$at each number to determine if $\left(x-1\right)\left(x+1\right)\left(x^2+3\right)$is positive or negative.

Notice, the function $\left(x-1\right)\left(x+1\right)\left(x^2+3\right)$is negative in the interval $(-1,1)$

Since we are looking where the function $\left(x-1\right)\left(x+1\right)\left(x^2+3\right)$is less than or equal to zero, the solution is:

$\left[-1,1\right]$

The solution of the inequality is $\left[-1,1\right]$

Use the graphing calculator, to graph the given function.

The graph is shown below:

The function $f\left(x\right)$is below $g\left(x\right)$in the interval $[-1,1]$.