Question

Find the critical numbers of the function.

32.\(g(x) = {x^{\frac{1}{3}}} - {x^{ - \frac{2}{3}}}\).

Short answer

\(x = - 2\)is the critical number of the function \(g(x) = {x^{\frac{1}{3}}} - {x^{ - \frac{2}{3}}}\).

Step-by-step solution

Given data

The function is \(g(x) = {x^{\frac{1}{3}}} - {x^{ - \frac{2}{3}}}\).

Concept of critical number

A critical number of a function\(f\)is a number\(c\), if it satisfies either of the below conditions:

(1)\({f^\prime }(c) = 0\)

(2)\({f^\prime }(c)\), Does not exist.

Obtain the first derivative of the given function

Obtain the first derivative of the given function \(g(x)\).

\({g^\prime }(x) = \frac{d}{{dt}}\left( {{x^{\frac{1}{3}}} - {x^{ - \frac{2}{3}}}} \right)\)

Apply the power rule:

\(\begin{aligned}{c}{g^\prime }(x) &= \frac{1}{3}{x^{\frac{1}{3} - 1}} - \left( { - \frac{2}{3}} \right){x^{ - \frac{2}{3} - 1}}\\ &= \frac{1}{3}{x^{ - \frac{2}{3}}} + \frac{2}{3}{x^{ - \frac{5}{3}}}\\ &= \frac{1}{{3{x^{\frac{2}{3}}}}} + \frac{2}{{3{x^{\frac{5}{3}}}}}\end{aligned}\)

Obtain the critical numbers

Set \({g^\prime }(x) = 0\)and obtain the critical numbers.

\(\begin{aligned}{c}\frac{{{x^{\frac{5}{3}}} + 2{x^{\frac{2}{3}}}}}{{3{x^{\frac{5}{3}}}{x^{\frac{2}{3}}}}} &= 0\\{x^{\frac{5}{3}}} + 2{x^{\frac{2}{3}}} &= 0\\{x^{\frac{2}{3}}}(x + 2) &= 0\end{aligned}\)

\( \Rightarrow x = - 2\;and\;x = 0\)

Note that, \(x = - 2\) do not satisfy the given equation.

Thus, \(x = 0\)is the critical number of the function \(g(x) = {x^{\frac{1}{3}}} - {x^{ - \frac{2}{3}}}\).