Question

Find the critical numbers of the function.

47. \(g\left( x \right) = {x^{\bf{2}}}{\bf{ln}}x\)

Short answer

The critical number is \(x = \frac{1}{{\sqrt e }}\).

Step-by-step solution

Critical Numbers of a function

TheCritical numbers for any function \(p\left( t \right)\) can be obtained by putting \(p'\left( t \right) = 0\).

Differentiating the function for critical numbers:

The given function is:

\(g\left( x \right) = {x^2}\ln x\)

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

\(\begin{aligned}{c}g'\left( x \right) &= \frac{d}{{dx}}\left( {{x^2}\ln x} \right)\\ &= {x^2}\frac{d}{{dx}}\left( {\ln x} \right) + \ln x\frac{d}{{dx}}\left( {{x^2}} \right)\\ &= {x^2}\frac{1}{x} + \ln x\left( {2x} \right)\\ &= x\left( {1 + 2\ln x} \right)\end{aligned}\)

Solving for critical numbers:

Solve further,

\(\begin{aligned}{c}x &= {e^{ - \frac{1}{2}}}\\x &= \frac{1}{{\sqrt e }}\end{aligned}\)

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