Question

Find the critical numbers of the function.

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

Short answer

The only critical number is \(x = - \frac{1}{6}\) .

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:

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

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

\(\begin{array}{c}f'\left( x \right) = \frac{d}{{dx}}\left( {3{x^2} + x - 2} \right)\\ = 6x + 1\end{array}\)

Now, solving for critical numbers:

\(\begin{array}{c}f'\left( x \right) = 0\\6x + 1 = 0\\x = - \frac{1}{6}\end{array}\)

Hence, the only critical number is \(x = - \frac{1}{6}\) .