Question

Find the critical numbers of the function.

40. \(g\left( x \right) = \sqrt(3){{4 - {x^2}}}\)

Short answer

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

Step-by-step solution

Critical Numbers of a function

TheCritical numbers for any function \(g\left( x \right)\) are obtained by putting \(g'\left( x \right) = 0\).

Differentiating the function for critical numbers:

The given function is:

\(g\left( x \right) = \sqrt(3){{4 - {x^2}}}\)

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

\(\begin{array}{c}g'\left( x \right) = \frac{d}{{dx}}\left( {\sqrt(3){{4 - {x^2}}}} \right)\\ = \frac{d}{{dx}}{\left( {4 - {x^2}} \right)^{\frac{1}{3}}}\\ = \frac{1}{3}{\left( {4 - {x^2}} \right)^{ - \frac{2}{3}}}\left( { - 2x} \right)\end{array}\)

Solving for critical numbers:

\(\begin{array}{c}g'\left( x \right) = 0\\\frac{1}{3}{\left( {4 - {x^2}} \right)^{ - \frac{2}{3}}}\left( { - 2x} \right) = 0\\x = 0\\x = 0\end{array}\)

Or,

\(\begin{array}{c}\left( {4 - {x^2}} \right) = 0\\x = \pm 2\end{array}\)

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