Question

graph the function in a viewing window that shows all of its extrema and x - intercepts. describe the end behavior using limits

f(x)=2x^{4}-5x^{3}-15x^{2}+13x + 45

choose the correct graph below.

a.
b.
c.
d.

Explanation:

Step1: Determine end - behavior

For the polynomial function \(f(x)=2x^{4}-5x^{3}-15x^{2}+13x + 45\), since the leading term is \(2x^{4}\) (even - degree and positive leading coefficient), \(\lim_{x
ightarrow\pm\infty}f(x)=+\infty\).

Step2: Find x - intercepts

We can try to find the roots of \(f(x)=2x^{4}-5x^{3}-15x^{2}+13x + 45\) by using the Rational Root Theorem. The possible rational roots are factors of \(\frac{45}{2}\), i.e., \(\pm1,\pm3,\pm5,\pm9,\pm15,\pm45,\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{5}{2},\pm\frac{9}{2},\pm\frac{15}{2},\pm\frac{45}{2}\). By testing \(x = - 1\), we have \(f(-1)=2+5 - 15 - 13 + 45=24
eq0\). By testing \(x = 3\), \(f(3)=2\times81-5\times27-15\times9 + 13\times3+45=162-135 - 135+39 + 45= - 24
eq0\). By testing \(x=- \frac{3}{2}\), \(f(-\frac{3}{2})=2\times\frac{81}{16}-5\times(-\frac{27}{8})-15\times\frac{9}{4}+13\times(-\frac{3}{2}) + 45=\frac{81}{8}+\frac{135}{8}-\frac{135}{4}-\frac{39}{2}+45=\frac{81 + 135-270 - 156+360}{8}=\frac{471 - 426}{8}=\frac{45}{8}
eq0\). By testing \(x = 5\), \(f(5)=2\times625-5\times125-15\times25+13\times5 + 45=1250-625-375 + 65+45=360
eq0\). Another way is to use a graphing utility or software to approximate the roots.

Step3: Find extrema

First, find the derivative \(f^\prime(x)=8x^{3}-15x^{2}-30x + 13\). We can use a graphing utility to find the critical points (where \(f^\prime(x)=0\)).
Since the end - behavior is \(\lim_{x
ightarrow\pm\infty}f(x)=+\infty\), we can eliminate graphs that have different end - behavior.

Answer:

We need the actual graphs to choose from, but based on the end - behavior \(\lim_{x
ightarrow\pm\infty}f(x)=+\infty\), we look for a graph that goes up on both the left - hand side (\(x
ightarrow-\infty\)) and the right - hand side (\(x
ightarrow+\infty\)). Without seeing the actual details of the graphs A, B, C, D, we can't give a definite letter - choice answer. But if we had to make a general statement, we would choose the graph that has the correct end - behavior and approximately the correct number of \(x\) - intercepts and extrema. If we assume that the end - behavior is the main differentiator among the options, we look for a graph that rises to the left and rises to the right.