Question

Find the critical numbers of the function.

\(f\left( x \right) = {\bf{3}}{x^{\bf{4}}} + {\bf{8}}{x^{\bf{3}}} - {\bf{48}}{x^{\bf{2}}}\)

Short answer

The critical numbers are \(x = - 4,{\rm{ }}0,{\rm{ and }}2\).

Step-by-step solution

Critical Numbers of a function

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

Differentiating the function for critical numbers

The given function is:

\(f\left( x \right) = 3{x^4} + 8{x^3} - 48{x^2}\)

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

\(\begin{array}{c}f'\left( x \right) = \frac{d}{{dx}}\left( {3{x^4} + 8{x^3} - 48{x^2}} \right)\\ = 12{x^3} + 24{x^2} - 96x\\ = 12x\left( {{x^2} + 2x - 8} \right)\\ = 12x\left( {x + 4} \right)\left( {x - 2} \right)\end{array}\)

Now, solving for critical numbers:

\(\begin{array}{c}f'\left( x \right) = 0\\12x\left( {x + 4} \right)\left( {x - 2} \right) = 0\\x = - 4,{\rm{ }}0,{\rm{ }}2\end{array}\)

Hence, the critical numbers are \(x = - 4,{\rm{ }}0,{\rm{ and }}2\).