Question

Find the critical numbers of the function.

\(p\left( t \right) = t{e^{{\bf{4}}t}}\)

Short answer

The critical number is \(t = - \frac{1}{4}\).

Step-by-step solution

Critical Numbers of a function

The Critical numbersfor 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:

\(p\left( t \right) = t{e^{4t}}\)

On differentiating with respect to\(t\);

\(\begin{aligned}{c}p'\left( t \right) &= \frac{d}{{dt}}\left( {t{e^{4t}}} \right)\\ &= t\frac{d}{{dt}}\left( {{e^{4t}}} \right) + {e^{4t}}\frac{d}{{dt}}\left( t \right)\\ &= 4t{e^{4t}} + {e^{4t}}\\ &= {e^{4t}}\left( {4t + 1} \right)\end{aligned}\)

Solving for critical numbers:

Thus, the only critical number is \(t = - \frac{1}{4}\).