Question

Find the critical numbers of the function.

30. \(g\left( v \right) = {v^{\bf{3}}} - {\bf{12}}v + {\bf{4}}\)

Short answer

The critical numbers are \(v = \pm 2\) .

Step-by-step solution

Critical Numbers of a function

TheCritical numbers for any function \(g\left( v \right)\) can be obtained by putting \(g'\left( v \right) = 0\).

Differentiating the function for critical numbers:

The given function is:

\(g\left( v \right) = {v^3} - 12v + 4\)

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

\(\begin{array}{c}g'\left( v \right) = \frac{d}{{dv}}\left( {{v^3} - 12v + 4} \right)\\ = 3{v^2} - 12\end{array}\)

Now, solving for critical numbers:

\(\begin{array}{c}g'\left( v \right) = 0\\3{v^2} - 12 = 0\\{v^2} = \frac{{12}}{3}\\v = \pm 2\end{array}\)

Hence, the critical numbers are \(v = \pm 2\).