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.

$f(x)=2x^{4}-16x^{3}$

choose the graph of the function.

a.
b.
c.
d.

Explanation:

Step1: Find domain

All real numbers (polynomial function).

Step2: Find intercepts

Set $f(x)=0$: $2x^3(x-8)=0 \Rightarrow x=0,8$. Y-intercept: $f(0)=0$.

Step3: Find critical points

$f'(x)=8x^3 - 48x^2 = 8x^2(x-6)$. Critical points at $x=0,6$.

Step4: Analyze increasing/decreasing

• $x<6$ (x≠0): $f'(x)≤0$ (decreasing)
• $x>6$: $f'(x)>0$ (increasing)

Minimum at $x=6$: $f(6)=2(1296)-16(216)=2592-3456=-864$.

Step5: Find inflection points

$f''(x)=24x^2 - 96x = 24x(x-4)$. Inflection at $x=0,4$.

Step6: Concavity

• $x<0$ or $x>4$: $f''(x)>0$ (concave up)
• $0

Step7: End behavior

As $x→±∞$, $2x^4$ dominates → $f(x)→+∞$.

Answer:

B. (Graph with minimum ~-864, concave up outside 0-4, concave down inside 0-4, intercepts at 0 and 8)