Chapter 4.2, Problem 21E

a.

To determine

To find: the local extrema of the given function.

Answer to Problem 21E

The given function has no local extrema.

Explanation of Solution

Given:

  $y=4-\sqrt{x+2}$ .

Calculation:

Consider the given function,

  $y=4-\sqrt{x+2}$

Differentiating with respect to $x$ , it follows that

  $\begin{array}{c}\frac{dy}{dx}=0-\frac{1}{2\sqrt{x+2}}\\ =-\frac{1}{2\sqrt{x+2}}\end{array}$

Here, the derivative cannot be zero for any real value of $x$ .

The derivative is not defined at $x=-2$ .

So, the critical value is $x=-2$ .

But the domain of the given function is $\left(-2,\infty \right)$ .

Hence the given function has no local extrema.

b.

To determine

To find: the intervals in which the given function is increasing.

Answer to Problem 21E

There is no interval in which the given function increasing.

Explanation of Solution

Given:

  $y=4-\sqrt{x+2}$ .

Concept used:

The function increases in the interval at which the first derivative is positive.

Calculation:

Consider the given function,

  $y=4-\sqrt{x+2}$

Differentiating with respect to $x$ , it follows that

  $\begin{array}{c}\frac{dy}{dx}=0-\frac{1}{2\sqrt{x+2}}\\ =-\frac{1}{2\sqrt{x+2}}\end{array}$

Now the function increases in the interval for which,

  $\begin{array}{c}\frac{dy}{dx}>0\\ -\frac{1}{2\sqrt{x+2}}>0\\ \frac{1}{\sqrt{x+2}}<0\end{array}$

It is not possible for any real value of $x$ .

Hence there is no interval in which the given function increasing.

c.

To determine

To find: the intervals in which the given function is decreasing.

Answer to Problem 21E

The given function decreases in the interval $\left(-2,\infty \right)$ .

Explanation of Solution

Given:

  $y=4-\sqrt{x+2}$ .

Concept used:

The function decreases in the interval at which the first derivative is negative.

Calculation:

Consider the given function,

  $y=4-\sqrt{x+2}$

Differentiating with respect to $x$ , it follows that

  $\begin{array}{c}\frac{dy}{dx}=0-\frac{1}{2\sqrt{x+2}}\\ =-\frac{1}{2\sqrt{x+2}}\end{array}$

Now the function decreases in the interval for which,

  $\begin{array}{c}\frac{dy}{dx}<0\\ -\frac{1}{2\sqrt{x+2}}<0\\ \frac{1}{\sqrt{x+2}}>0\\ x>-2\end{array}$

Hence the given function decreases in the interval $\left(-2,\infty \right)$ .