Question

In Problems 95–102, analyze each polynomial function by following Steps 1 through 8 on page 190.

[Hint: You will need to first factor the polynomial.]

$f\left(x\right)=-x^5+5x^4+4x^3-20x^2$

Short answer

1. The graph of f has an end behavior like$y=-x^5$

Step-by-step solution

Step 1. Given Information

$f\left(x\right)=-x^5+5x^4+4x^3-20x^2$

Step 2. Factorization

$$\begin{gathered} f\left(x\right)=-x^5+5x^4+4x^3-20x^2 \\ =-x^2(x^3-5x^2-4x+20) \\ =-x^2\left\{x^2\right(x-2)--3x(x-2)-10(x-2\left)\right\} \end{gathered}$$

localid="1650169764787" $$\begin{gathered} =-x^2(x-2)\{x^2-3x-10\} \\ =-x^2(x-2)\{x^2-5x+2x-10\} \\ =-x^2(x-2)\left\{x\right(x-5)+2(x-5\left)\right\} \\ =-x^2(x-2)(x-5)(x+2) \end{gathered}$$

Step 3. End Behavior

$f\left(x\right)=-x^5+5x^4+4x^3-20x^2$

Clearly, as $x\to \infty ,f\left(x\right)\to -\infty$

and as$x\to -\infty ,f\left(x\right)\to \infty$

Step 4. Graph

Step 5. x and y intercept

For $y-$intercept

$f\left(0\right)=0$i.e.$(0,0)$

and for $x-$intercept

$f\left(x\right)=0$

or, $-x^2(x-2)(x-5)(x+2)=0$

Then $x=-2,0,2,5$

Then $x-$intercepts are$(-2,0),(0,0),(2,0),(5,0)$

Step 6. Calculation

Now, we have to find points on the function around x & y intercept

For $x=-2$

$x$	$y$
$-3$	$360$
$-1$	$-18$

For$x=0$

$x$	$y$
$-1$	$-18$
$1$	$-12$

For $x=2$

$x$	$y$
$3$	$90$
$1$	$-12$

For $x=5$

$x$	$y$
$6$	$-1152$
$4$	$192$

Step 7. Turning Points

The turning points of the function in the graph are$(-1.451,-25.732),(0,0),(1.341,-14.487),(4.11,193.89)$

Step 8. Graph

Step 9. Domain and Rnage

The domain and the range of the given function is the set of all real numbers.

Step 10. Increasing and Decreasing points

The function is increasing on $-1.451<x\le 0\&1.341<x\le 4.11$.

The function is decreasing on$-\infty <x\le -1.451,0<x\le 1.341\&4.11<x<\infty$.