Question

How do you find the critical numbers of \[f\left( x \right)={{x}^{\dfrac{1}{5}}}-{{x}^{-\dfrac{4}{5}}}\]?

Answer 1

Hint: The critical numbers of a function are the numbers belonging to the domain of the function and are defined with respect to the derivative of the function. All the values of the independent variable for which the derivative of the function is equal to zero, or is not defined are termed as the critical numbers. So in order to get the critical numbers of the function given in the question, we need to differentiate the function and equate the resulting derivative and its denominator to zero. On solving the equations thus obtained, we will get all the critical numbers of the given function.

Complete step by step answer:
The function given in the question is
\[f\left( x \right)={{x}^{\dfrac{1}{5}}}-{{x}^{-\dfrac{4}{5}}}\]
The graph of the above function is

We know that the differentiation of the function ${{x}^{n}}$ is equal to $n{{x}^{n-1}}$. Applying this, we differentiate the above function to get the derivative of the given function as
\[\begin{align}
  & \Rightarrow f'\left( x \right)=\dfrac{1}{5}{{x}^{\dfrac{1}{5}-1}}-\left( -\dfrac{4}{5} \right){{x}^{-\dfrac{4}{5}-1}} \\
 & \Rightarrow f'\left( x \right)=\dfrac{1}{5}{{x}^{-\dfrac{4}{5}}}+\dfrac{4}{5}{{x}^{-\dfrac{9}{5}}} \\
\end{align}\]
Taking $\dfrac{1}{5}{{x}^{-\dfrac{9}{5}}}$ common, we get the simplified expression for the derivative of the function as
\[\begin{align}
  & \Rightarrow f'\left( x \right)=\dfrac{1}{5}{{x}^{-\dfrac{9}{5}}}\left( {{x}^{-\dfrac{4}{5}+\dfrac{9}{5}}}+4{{x}^{-\dfrac{9}{5}+\dfrac{9}{5}}} \right) \\
 & \Rightarrow f'\left( x \right)=\dfrac{1}{5}{{x}^{-\dfrac{9}{5}}}\left( x+4 \right) \\
\end{align}\]
We know that ${{x}^{-n}}=\dfrac{1}{{{x}^{n}}}$. Using this we can write the above expression for the derivative as
\[\Rightarrow f'\left( x \right)=\dfrac{1}{5}\dfrac{\left( x+4 \right)}{{{x}^{\dfrac{9}{5}}}}.........(i)\]
Now, for the critical numbers, we equate the derivative of the function to zero to get
\[\Rightarrow \dfrac{1}{5}\dfrac{\left( x+4 \right)}{{{x}^{\dfrac{9}{5}}}}=0\]
Multiplying by \[5{{x}^{\dfrac{9}{5}}}\] both the sides, we get
\[\Rightarrow x+4=0\]
Subtracting \[4\] from both the sides, we get
$\begin{align}
  & \Rightarrow x+4-4=0-4 \\
 & \Rightarrow x=-4 \\
\end{align}$
Also, we equate the denominator of (i) to zero to get
\[\begin{align}
  & \Rightarrow {{x}^{\dfrac{9}{5}}}=0 \\
 & \Rightarrow x=0 \\
\end{align}\]
Hence, the critical numbers of the given function are \[0\] and \[-4\].

Note:
We must remember that the critical numbers are the points on the graph of the function where its derivative is equal to zero or it is not defined. So do not forget to equate the denominator of the derivative of the function to zero, after equating the derivative to zero.