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

Domain is all real numbers ($\mathbb{R}$).

Step2: Find intercepts

Set $f(x)=0$: $2x^3(x-8)=0$ → $x=0$ or $x=8$. Y-intercept: $f(0)=0$.

Step3: Find critical points

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

Step4: Test 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 concavity/inflection

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

• $x<0$ or $x>4$: concave up
• $0

Step6: Match graph

Graph has minimum at $x=6$ (≈-864), intercepts at 0 and 8, concave down between 0 and 4.

Answer:

B (assuming option B matches the above characteristics)