Question

In Problems 49– 60, for polynomial function

$f\left(x\right)=-2x^2\left(x^2-2\right)$

(a) List each real zero and its multiplicity.

(b) Determine whether the graph crosses or touches the x-axis at each x-intercept.

(c) Determine the maximum number of turning points on the graph.

(d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of $\left|x\right|$.

Short answer

1. Zeros of the function are $0$with multiplicity 2 and $\pm \sqrt{2}$with multiplicity 1.
2. The graph of the function touches at $x=0$and crosses x-axis at $x=\pm \sqrt{2}$
3. The turning points are 3.
4. The graph of function behave like graph of $-2x^4$for large values of$\left|x\right|$.

Step-by-step solution

Step 1. Given Information         

The function is

$f\left(x\right)=-2x^2\left(x^2-2\right)$

Part (a) Step 1. To find zeros and their multiplicity.      

Equate given function with zero.

$$\begin{gathered} f\left(x\right)=0 \\ -2x^2\left(x^2-2\right)=0 \end{gathered}$$

when

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

So this has multiplicity 2.

When

$$\begin{gathered} x^2-2=0 \\ {x}^2=2 \\ x=\pm \sqrt{2} \end{gathered}$$

it has multiplicity 1

Part (b) Step 1. To determine whether the graph crosses or touches the x-axis at each x-intercept.       

x-intercepts for the function is 0 with multiplicity 2 so graph of the function touches x-axis.

and $\pm \sqrt{2}$with multiplicity 1 so the graph will cross the x-axis at these points.

Part (C) Step 1. To determine the maximum number of turning points on the graph       

Expand the given function

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

Here we can see that polynomial has degree 4

So turning point of the graph is $4-1=3$

Part (d). Step 1. To find end behavior of graph        

Since the highest power of $x$after expanding the function is 4.

So the graph of the polynomial will behave like graph of$-2x^4$

for large values of $\left|x\right|$.