Question

Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1

$f\left(x\right)=6x^4-37x^3+58x^2+3x-18$

Short answer

For the given polynomial function, we get:

x-intercept$=-\frac{1}{2},\frac{2}{3},3$

y-intercept$=-18$

Step-by-step solution

Step 1. End behavior of the function

We have a degree of five from$f\left(x\right)=6x^4-37x^3+58x^2+3x-18$

Thus, the graph of function has an end behavior similar to $y=6x^4$

Step 2. Determine the intercept of the function

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

$\begin{array}{rcrrr}6x^4-37x^3+58x^2+3x-18& =0& & & \\ (2x+1)(3x-2)(x-3{)}^2& =0& & & \\ 2x+1=0;& 3x-2& =0;& x-3& =0\\ 2x=-1;& 3x& =2;& x& =3\\ x=-\frac{1}{2};& x=\frac{2}{3}& & & \end{array}$

Considering$f\left(0\right)=-18$, thus y-intercept is $(0,-18)$.

Step 3. Graph the function

Step 4. Maximum and minimum turning point

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

$(1.65,23.118)$

Minimum:$(3,0)$and$(-0.025,-18.038)$

Step 5. Graph of function with obtained result

Step 6. Domain and range

Therefore, the domain of the function is the set of all real numbers and range is$(-18.038,\infty )$

Step 7. Interval at which the graph is increasing or decreasing

By observing the graph above, we say that the function at the intervals of

$(-0.025,1.65),(3,\infty )$is increasing

At intervals of$(-\infty ,-0.025),(1.65,3)$is decreasing.