Question

graph the function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave upward or concave downward, intercepts where possible, and asymptotes where applicable.

$f(x)=-4x^{3}-12x^{2}+180x - 11$

choose the graph of the function.

a.
b.
c.
d.

Explanation:

Step1: Find domain

Domain is all real numbers ($\mathbb{R}$) since it's a polynomial.

Step2: Compute first derivative

$f'(x) = -12x^2 - 24x + 180$

Step3: Find critical points

Set $f'(x)=0$:
$-12(x^2 + 2x - 15)=0 \Rightarrow (x+5)(x-3)=0 \Rightarrow x=-5, x=3$

Step4: Determine increasing/decreasing

• $f'(x)>0$ when $-5
• $f'(x)<0$ when $x<-5$ or $x>3$ (decreasing)

Step5: Compute second derivative

$f''(x) = -24x -24$

Step6: Find inflection point

Set $f''(x)=0$:
$-24(x+1)=0 \Rightarrow x=-1$

Step7: Check concavity

• $f''(x)>0$ when $x<-1$ (concave up)
• $f''(x)<0$ when $x>-1$ (concave down)

Step8: Analyze end behavior

As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$.

Step9: Match with graphs

The function has critical points at $x=-5,3$, inflection at $x=-1$, increasing between $-5$ and $3$, and end behavior $x\to\infty$: $f(x)\to-\infty$. This matches graph A.

Answer:

A. [Graph A]