which best describes the graph of the cubic function (f(x)=x^{3}+x^{2}+x + 1)?\nas (x) increases, (y)…

Explanation:

Step1: Find the derivative

Differentiate $f(x)=x^{3}+x^{2}+x + 1$ using the power - rule. The derivative $f^\prime(x)=3x^{2}+2x + 1$.

Step2: Analyze the discriminant of the quadratic

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

Step3: Determine the behavior of the function

Since $f^\prime(x)>0$ for all real $x$, the function $y = f(x)$ is increasing for all real values of $x$. That is, as $x$ increases, $y$ increases along the entire graph.

Answer:

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