Question

Find the critical numbers of the function.

48. \(B\left( u \right) = {\bf{4ta}}{{\bf{n}}^{ - {\bf{1}}}}u - u\)

Short answer

The critical numbers are \(u = \pm \sqrt 3 \).

Step-by-step solution

Critical Numbers of a function

The Critical 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:

\(B\left( u \right) = 4{\tan ^{ - 1}}u - u\)

On differentiating, we have:

\(\begin{aligned}{c}B'\left( u \right) &= \frac{d}{{du}}\left( {4{{\tan }^{ - 1}}u - u} \right)\\ &= 4\frac{d}{{du}}\left( {{{\tan }^{ - 1}}u} \right) - \frac{d}{{du}}\left( u \right)\\ &= 4\left( {\frac{1}{{1 + {u^2}}}} \right) - 1\\ &= \frac{{3 - {u^2}}}{{1 + {u^2}}}\end{aligned}\)

Solving for critical numbers:

Hence, the critical numbers obtained are \(u = \pm \sqrt 3 \).