which best describes the graph of the cubic function f(x) = x³ + x² + x + 1?\nas x increases, y increases…

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=x^{3}+x^{2}+x + 1$ is $f^\prime(x)=3x^{2}+2x + 1$.

Step2: Analyze the discriminant of the quadratic - derivative

For a quadratic function $ax^{2}+bx + c$ (here $a = 3$, $b = 2$, $c = 1$), the discriminant $\Delta=b^{2}-4ac$. Substitute the values: $\Delta=(2)^{2}-4\times3\times1=4 - 12=-8<0$. And since $a = 3>0$, the quadratic function $y = f^\prime(x)=3x^{2}+2x + 1>0$ for all real - valued $x$.

Step3: Determine the monotonicity of the original function

If $f^\prime(x)>0$ for all $x$ in the domain of $f(x)$ (which is $(-\infty,\infty)$), then the function $y = f(x)$ is increasing on the entire real line. That is, as $x$ increases, $y$ increases along the entire graph.

Answer:

As $x$ increases, $y$ increases along the entire graph.