Question

43.Find the critical numbers of the function.

43. \(f\left( x \right) = {x^{\frac{{\bf{1}}}{{\bf{3}}}}}{\left( {{\bf{4}} - x} \right)^{\frac{{\bf{2}}}{{\bf{3}}}}}\)

Short answer

The critical numbers are \(x = 0,\frac{4}{3}{\rm{ and }}4\).

Step-by-step solution

Critical Numbers of a function

TheCritical numbers for any function \(f\left( x \right)\) are obtained by putting \(f'\left( x \right) = 0\).

Differentiating the function for critical numbers:

The given function is:

\(f\left( x \right) = {x^{\frac{1}{3}}}{\left( {4 - x} \right)^{\frac{2}{3}}}\)

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

\(\begin{aligned}{c}f'\left( x \right) &= \frac{d}{{dx}}\left( {{x^{\frac{1}{3}}}{{\left( {4 - x} \right)}^{\frac{2}{3}}}} \right)\\ &= {x^{\frac{1}{3}}}\frac{d}{{dx}}{\left( {4 - x} \right)^{\frac{2}{3}}} + {\left( {4 - x} \right)^{\frac{2}{3}}}\frac{d}{{dx}}\left( {{x^{\frac{1}{3}}}} \right)\\ &= \frac{2}{3}{x^{\frac{1}{3}}}{\left( {4 - x} \right)^{ - \frac{1}{3}}} + \frac{1}{3}{\left( {4 - x} \right)^{\frac{2}{3}}}{x^{ - \frac{2}{3}}}\\ &= \frac{1}{3}{x^{ - \frac{2}{3}}}{\left( {4 - x} \right)^{ - \frac{1}{3}}}\left\{ {4 - 3x} \right\}\end{aligned}\)

Solving for critical numbers:

\(\begin{aligned}{c}f'\left( x \right) &= 0\\\frac{1}{3}{x^{ - \frac{2}{3}}}{\left( {4 - x} \right)^{ - \frac{1}{3}}}\left\{ {4 - 3x} \right\} &= 0\\x &= 0,\frac{4}{3}{\rm{ and 4}}\end{aligned}\)

Hence, the critical numbers are \(x = 0,\frac{4}{3}{\rm{ and }}4\).