Question

Find the critical numbers of the function.

32. \(f\left( x \right) = {\bf{2}}{x^{\bf{3}}} + {x^{\bf{2}}} + {\bf{8}}x\)

Short answer

There are no critical numbers.

Step-by-step solution

Critical Numbers of a function

TheCritical numbersfor any function \(f\left( x \right)\) are obtained by putting \(f'\left( x \right) = 0\).

Differentiating the function for critical numbers:

The given function is:

\(f\left( x \right) = 2{x^3} + {x^2} + 8x\)

On differentiating with respect to\(x\)as:

\(\begin{array}{c}f'\left( x \right) = \frac{d}{{dx}}\left( {2{x^3} + {x^2} + 8x} \right)\\ = 6{x^2} + 2x + 8\end{array}\)

Now, solving for critical numbers:

\(\begin{array}{c}f'\left( x \right) = 0\\6{x^2} + 2x + 8 = 0\\x = \frac{{ - 1 \pm \sqrt { - 47} }}{6}\end{array}\)

Here, \(\sqrt { - 47} \) is an imaginary value.

Hence, there is no critical number obtained.