Chapter 1.3, Problem 36E

Solution

a.

To Determine: The interval in which the function is increasing or decreasing.

The function is increasing on the interval of $\left(10,\infty \right)$ .

Given information:

  $r\left(x\right)=\sqrt{x-10}$

Graph:

  

From the graph, it can be seen that the graph is increasing on the interval $\left(10,\infty \right)$ .

b.

To Determine: The function is odd, even or neither.

The function is neither even nor odd.

Given information:

  $r\left(x\right)=\sqrt{x-10}$

Calculation:

A function is even if $f\left(-x\right)=f\left(x\right)$

Check if $f\left(-x\right)=f\left(x\right)$ :

  $\sqrt{-x-10}\ne \sqrt{x-10}$

Since $\sqrt{-x-10}\ne \sqrt{x-10}$ , the function is not even.

A function is odd if $f\left(-x\right)=-f\left(x\right)$

Multiply $-1$ by $\sqrt{x-10}$ :

  $-f\left(x\right)=-\sqrt{x-10}$

Since $\sqrt{-x-10}\ne -\sqrt{x-10}$ , the function is not odd.

So, the function is neither even nor odd.

c.

To Determine: The extrema of the function, if any.

There is no extrema points.

Given information:

  $r\left(x\right)=\sqrt{x-10}$

Calculation:

Find the first derivative of the function:

  $r^{\text{'}}\left(x\right)=-\frac{1}{4{(x-10)}^{\frac{3}{2}}}\frac{1}{2{\left(x-10\right)}^{\frac{1}{2}}}$

Find the second derivative of the function.

  $r^{\prime \prime }\left(x\right)=-\frac{1}{4{(x-10)}^{\frac{3}{2}}}$

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

  $\frac{1}{2{(x-10)}^{\frac{1}{2}}}=0$

Set the first derivative equal to $0$ then solve the equation $\frac{1}{2{(x-10)}^{\frac{1}{2}}}=0$

  $\frac{1}{2{(x-10)}^{\frac{1}{2}}}=0$

Set the numerator equal to $0$ :

  $1=0$

Since $1\ne 0$ there is no solution.

Convert expressions with fractional exponents to radicals:

  $\frac{1}{2\sqrt{x-10}}$

Set the denominator in $\frac{1}{2\sqrt{x-10}}$ equal to $0$ to find where the expression is undefined.

  $2\sqrt{x-10}=0$

Solve for $x:$

  $\begin{array}{c}{(}^2=0^2\\ 4x-40=0\\ x=10\end{array}$

Find the values where the derivative is undefined.

  $\begin{array}{l}x\le 10\\ \left(-\infty ,10\right]\end{array}$

Critical points to evaluate.

  $x=10$

Evaluate the second derivative at $x=10$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

  $\begin{array}{c}=-\frac{1}{4{((10)-10)}^{\frac{3}{2}}}\\ =-\frac{1}{4\cdot 0^3}\\ =-\frac{1}{0}\end{array}$

The expression is undefined. So, there is no extrema points plotted in the graph.

d.

To Determine: The graph of the function related to a graph of one of the twelve basic functions.

The graph is related to square root function.

Given information:

  $r\left(x\right)=\sqrt{x-10}$

Calculation:

  

The function $r\left(x\right)=\sqrt{x-10}$ is related to square root function. As its argument is $x-10$ , it is shifted $10$ units to the right.