Question

cubic and cube root functions and equations
which is the graph of the function $f(x)=x^3+x^2+x+1$?

Explanation:

Step1: Analyze end behavior

For cubic $f(x)=x^3+x^2+x+1$, leading term $x^3$: as $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to-\infty$. Eliminates graphs with wrong end behavior.

Step2: Calculate y-intercept

Set $x=0$: $f(0)=0^3+0^2+0+1=1$. The graph crosses y-axis at $(0,1)$.

Step3: Check critical points

Find derivative: $f'(x)=3x^2+2x+1$. Discriminant: $\Delta=2^2-4(3)(1)=4-12=-8<0$. No real critical points, so function is always increasing.

Step4: Match to graph

Only the top graph has correct end behavior, y-intercept at $(0,1)$, and is strictly increasing.

Answer:

The top-most graph (Option C)