Question

Analyze each polynomial function.

$f\left(x\right)=4x^3+10x^2-4x-10$

Short answer

Domain and range: all real numbers

At intervals of $(-\infty ,-1.847)$and $(0.18,\infty )$is increasing; At intervals of$(-1.847,0.18)$is decreasing

Step-by-step solution

Given Information

The given function is:

$f\left(x\right)=4x^3+10x^2-4x-10$

Find the degree

Notice that, We have a degree of $3$from $f\left(x\right)=4x^3+10x^2-4x-10$

Thus, The graph of $f$ has an end behavior similar to$y=4x^3.$

Find the x intercept

Now let's determine the intercepts of $f$To get the $x$-intercept, equate $f\left(x\right)$to zero

factor out $2$

$4x^3+10x^2-4x-10=0$

factor out grouping

$$\begin{gathered} 2\left(2x^3+5x^2-2x-5\right)=0 \\ 2x(2x+5)(x+1)(x-1)=0 \end{gathered}$$

$$\begin{gathered} 2x=0 \\ x=0 \end{gathered}$$

or

$$\begin{gathered} 2x+5=0 \\ x=-\frac{5}{2} \end{gathered}$$

or

$$\begin{gathered} x+1=0 \\ x=-1 \end{gathered}$$

or

$$\begin{gathered} x-1=0 \\ x=1 \end{gathered}$$

So, the $x$-intercepts are $x=0,x=-\frac{5}{2},x=-1,x=1$

Considering $f\left(0\right)=-10,$thus the $y$-intercept is $(0,-10)$

Note that the zeros of $f$are: $0,-\frac{5}{2},-1,1$ with the multiplicity of 1 is a zero that has an odd multiplicity, Thus the graph crosses the $x$-axis

The graph

From the graph above, we can also see that the MAXIMUM turning point of the function is at point $(-1.847,6.299)$and the MINIMUM turning point is at point$(0.18,-10.373)$

Therefore, the function has a domain and range of all real numbers; or simply$(-\infty ,\infty )$

By observing the graph above We say that the function At intervals of $(-\infty ,-1.847)$and $(0.18,\infty )$is increasing; At intervals of $(-1.847,0.18)$is decreasing