Question

Find the critical numbers of the function.

33. \(g\left( t \right) = {t^{\bf{5}}} + {\bf{5}}{t^{\bf{3}}} + {\bf{50}}t\)

Short answer

There are no critical numbers.

Step-by-step solution

Critical Numbers of a function

TheCritical numbers for any function \(g\left( t \right)\) are obtained by putting \(g'\left( t \right) = 0\).

Differentiating the function for critical numbers:

The given function is:

\(g\left( t \right) = {t^5} + 5{t^3} + 50t\)

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

\(\begin{array}{c}g'\left( t \right) = \frac{d}{{dt}}\left( {{t^5} + 5{t^3} + 50t} \right)\\ = 5{t^4} + 15{t^2} + 50\\ = 5\left( {{t^4} + 3{t^2} + 10} \right)\end{array}\)

Now, solving for critical numbers:

\(\begin{array}{c}g'\left( t \right) = 0\\5\left( {{t^4} + 3{t^2} + 10} \right) = 0\\{t^2} = \frac{{ - 3 \pm \sqrt {{3^2} - 40} }}{2}\\t = \pm \sqrt {\frac{{ - 3 \pm \sqrt { - 31} }}{2}} \end{array}\)

Here, \(\sqrt { - 31} \) is not a real number.

Hence, the critical numbers for this function do not exist.