Single Variable Calculus, Chapter 4, 4.1, Section 4.1, Problem 32

Determine the critical numbers of the function $f(x) = x^3 + x^2 + x$


$
\begin{equation}
\begin{aligned}

f'(x) =& \frac{d}{dx} (x^3) + \frac{d}{dx} (x^2) + \frac{d}{dx} (x)
\\
\\
f'(x) =& 3x^2 + 2x + 1

\end{aligned}
\end{equation}
$


Solving for critical numbers


$
\begin{equation}
\begin{aligned}

f'(x) =& 0
\\
\\
0 =& 3x^2 + 2x + 1

\end{aligned}
\end{equation}
$


Using Quadratic Equation to solve for $x$


$
\begin{equation}
\begin{aligned}

x =& \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad ax^2 + bx + c = 0
\\
\\
x =& \frac{-2 \pm \sqrt{(2)^2 - (4)(3)(1)}}{2(3)}
\\
\\
x =& \frac{-2 \pm \sqrt{4 - 12}}{6}
\\
\\
x =& \frac{-2 \pm \sqrt{-8}}{6}

\end{aligned}
\end{equation}
$


The derivative of the function has no real solution.

Therefore, there are no critical numbers.