Question

Find the critical numbers of the function.

31.\(F(x) = {x^{\frac{4}{5}}}{(x - 4)^2}\).

Short answer

The critical number of the function \(F(x) = {x^{\frac{4}{5}}}{(x - 4)^2}\) are \(x = 0,\;\frac{8}{7}\), and \(4\).

Step-by-step solution

Given data

The function is \(F(x) = {x^{\frac{4}{5}}}{(x - 4)^2}\).

Concept of critical number and product rule

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.

The Product Rule: If two functions\(g(x)\)and\(h(x)\)are differentiable, then the derivative of\(f(x) = h(x)g(x)\)is,\({f^\prime }(x) = h(x){g^\prime }(x) + g(x){h^\prime }(x)\).

Obtain the first derivative of the given function

Obtain the first derivative of the given function, \(F'(x) = \frac{d}{{dx}}\left( {{x^{\frac{4}{5}}}{{(x - 4)}^2}} \right)\).

Apply the product rule:

\(\begin{aligned}{c}F'(x) &= {x^{\frac{4}{5}}}\frac{d}{{dx}}{(x - 4)^2} + {(x - 4)^2}\frac{d}{{dt}}\left( {{x^{\frac{4}{5}}}} \right)\\ &= {x^{\frac{4}{5}}} \cdot 2(x - 4) + {(x - 4)^2} \cdot \frac{4}{5}{x^{\frac{{ - 1}}{5}}}\\ &= \frac{1}{5}{x^{\frac{{ - 1}}{5}}}(x - 4)(5 \cdot 2 \cdot x + (x - 4) \cdot 4)\\ &= \frac{1}{5}{x^{\frac{{ - 1}}{5}}}(x - 4)(10x + 4x - 16)\end{aligned}\)

Simplify further as shown below.

\(\begin{aligned}{c}{F^\prime }(x) = \frac{1}{5}{x^{\frac{{ - 1}}{5}}}(x - 4)(14x - 16)\\ = \frac{{(x - 4)(14x - 16)}}{{5{x^{\frac{1}{5}}}}}\\ = \frac{{2(x - 4)(7x - 8)}}{{5{x^{\frac{1}{5}}}}}\end{aligned}\)

Obtain the critical number

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

\(\begin{aligned}{c}\frac{{2(x - 4)(7x - 8)}}{{5{x^{\frac{1}{5}}}}} &= 0\\2(x - 4)(7x - 8) &= 0\\(x - 4)(7x - 8) &= 0\end{aligned}\)

Therefore, the solutions are \(x = 4\) and \(x = \frac{8}{7}\).

The derivative function \({F^\prime }(x)\) does not exist when, \(x = 0\).

Hence, it satisfies the condition of the critical number definition.

Therefore, the critical number occurs at \(x = 0.\)

Thus, the critical numbers of the function \(F(x)\) are \(x = 0,\;\frac{8}{7}\), and \(4\).