Chapter 5.2, Problem 36E

To determine

To find : The function of the derivative $f\text{'}\left(x\right)=\frac{1}{4x^{3/4}}$ whose graph passes through the point $P\left(1,-2\right)$ .

Answer to Problem 36E

The required function is $f\left(x\right)=x^{1/4}-3$ .

Explanation of Solution

Given information :

The given derivative is $f\text{'}\left(x\right)=\frac{1}{4x^{3/4}}$ and the point is $P\left(1,-2\right)$ .

Calculation :

The function $F\left(x\right)$ is the antiderivative of the function $f\left(x\right)$ if $F\text{'}\left(x\right)=f\left(x\right)$ for all $x$ in the domain of $f$ .

Since, the function $f$ has the same derivative as the function $g\left(x\right)=x^{1/4}$ . Add a constant $C$ to $g\left(x\right)=x^{1/4}$ to get the function $f$ as the derivative of constant will always be zero.

Therefore, the antiderivative is $f\left(x\right)=x^{1/4}+C$ ...... (1)

Substitute the point $P\left(1,-2\right)$ in above function and solve for constant $C$ :

  $\begin{array}{l}-2=1^{1/4}+C\\ -2=1+C\\ C=-3\end{array}$

Now, substitute the value of $C$ in equation (1 )to get the required function:

  $f\left(x\right)=x^{1/4}-3$

Hence,

The required function is $f\left(x\right)=x^{1/4}-3$ .