Chapter 3, Problem 47RE

(a)

To determine

To find : The value of $f^{\prime }\left(x\right)$

Explanation of Solution

Given: The given function is $f\left(x\right)=x\sqrt{5-x}$ .

Consider the equation.

  $f\left(x\right)=x\sqrt{5-x}$

Differentiate the above equation.

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{d}{dx}\left(x\sqrt{5-x}\right)\\ =\sqrt{5-x}\cdot \left(1\right)+x\left(\frac{-1}{2}{\left(5-x\right)}^{-1/2}\right)\\ =\sqrt{5-x}+x\left(\frac{-1}{2}{\left(5-x\right)}^{-1/2}\right)\end{array}$

Therefore, the value of $f^{\prime }\left(x\right)$ is $\sqrt{5-x}+x\left(\frac{-1}{2}{\left(5-x\right)}^{-1/2}\right)$ .

(b)

To determine

To find : The equation of tangent to the curve at the point $\left(1,2\right)$ and $\left(4,4\right)$

Explanation of Solution

Given: The given function is $f\left(x\right)=x\sqrt{5-x}$ .

Consider the equation.

  $f\left(x\right)=x\sqrt{5-x}$

Differentiate the above equation.

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{d}{dx}\left(x\sqrt{5-x}\right)\\ =\sqrt{5-x}\cdot \left(1\right)+x\left(\frac{-1}{2}{\left(5-x\right)}^{-1/2}\right)\\ =\sqrt{5-x}+x\left(\frac{-1}{2}{\left(5-x\right)}^{-1/2}\right)\end{array}$

Substitute $1$ for $x$ to get the slope of tangent at $\left(1,2\right)$ .

  $\begin{array}{c}{y}^{\prime }\left(1\right)=\sqrt{5-1}-\frac{1}{2\sqrt{5-1}}\\ =2-\frac{1}{4}\\ =\frac{7}{4}\end{array}$

The equation of tangent at $\left(1,2\right)$ is,

  $\begin{array}{c}\frac{y-2}{x-1}=\frac{7}{4}\\ 4y-8=7x-7\\ 4y-7x-1=0\end{array}$

Substitute $4$ for $x$ to get the slope of tangent at $\left(4,4\right)$ .

  $\begin{array}{c}{y}^{\prime }\left(4\right)=\sqrt{5-4}-\frac{1}{2\sqrt{5-4}}\\ =1-\frac{4}{2}\\ =-1\end{array}$

The equation of tangent at $\left(4,4\right)$ is,

  $\begin{array}{c}\frac{y-4}{x-4}=-1\\ y-4=-x+4\\ y+x-8=0\end{array}$

Therefore, the equation of tangent is listed above.

(c)

To determine

To illustrate : The equation of tangent to the curve at the point $\left(1,2\right)$ and $\left(4,4\right)$ with the help of a graph.

Explanation of Solution

Given: The given function is $f\left(x\right)=x\sqrt{5-x}$ .

The graph of equation of tangent passing through $\left(1,2\right)\text{and}\left(4,4\right)$ is shown in figure below.

  

Figure (1)

Therefore, the equation of tangent passing through $\left(1,2\right)\text{and}\left(4,4\right)$ on the same plane is shown in Figure (1).

(d)

To determine

To check : whether the graph of $f$ and $f^{\prime }$ is reasonable.

Explanation of Solution

Given: The given function is $f\left(x\right)=x\sqrt{5-x}$ .

The graph of $f\left(x\right)$ and $f^{\prime }\left(x\right)$ is shown in figure below.

  

Figure (1)

Therefore, the original function is increasing the derivative function also remains positive. Hence, the result are reasonable.