91Ó°ÊÓ

We have a degree of five from $f\left(x\right)=4x^5+12x^4−x−3$.

Thus, the graph of function has an end behavior similar to $y=4x^5$.

Determine the intercept of function to get the x- intercept.

$\begin{array}{r}4x^5+12x^4−x−3â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =0\\ 4x^4(x+3)−(x+3)â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =0\\ \left(4x^4−1\right)(x+3)â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =0\\ 2x^2+1â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =0;\begin{array}{rlrlrl}\left(2x^2+1\right)\left(2x^2−1\right)(x+3)& =0& & & & \\ 2x^2−1& =0;x+3& =0& & & \\ 2x^2& =−1;& 2x^2& =1;& x& =−3\\ \sqrt{x^2}& =−\sqrt{\frac{1}{2};}& \sqrt{x^2}& =\sqrt{\frac{1}{2}}& & \\ x& =−\frac{\sqrt{2}}{2};& x& =\frac{\sqrt{2}}{2}& & \end{array}\end{array}$

Considering $f\left(0\right)=-3$we have y-intercept$(0,-3)$

From the graph above, we can see that the maximum and minimum turning point of the function is:

$(-2.404,79.028)$and$(0.266,-3.201)$

Therefore, the domain and range of the function is the set of all real numbers.

By observing the graph above, we say that the function at the intervals of $(-∞,2.404)$and $(0.266,∞)$is increasing.

At intervals of $(-2.404,0.266)$decreasing.