Q. 3 Intervals of behavior: For each ... [FREE SOLUTION]

Chapter 3: Q. 3 (page 314)

Intervals of behavior: For each of the following functions $f$ , determine the intervals on which $f$ is positive, negative, increasing, decreasing, concave up, and concave down.
$f (x) = \frac{x 鈭� 1}{x + 3}$

Short Answer

Expert verified

The $f^{'}$ always positive on $(- 鈭�, - 3) 鈭� (- 3, 鈭�)$ that is $f$ is always increasing $(- 鈭�, - 3) 鈭� (- 3, 鈭�)$ . The function is a concave up $(- 鈭�, - 3)$ and concave down at a point $(- 3, 鈭�)$ .

Step by step solution

Step 1. Given data

We have been given the function ,

$f (x) = \frac{x 鈭� 1}{x + 3}$

Step 2. Critical points

We have to find the derivative of the given function

$f (x) = \frac{x 鈭� 1}{x + 3}$

Therefore,

$\begin{aligned} f^{鈥�} (x) & = \frac{(x + 3) 鈰� 1 鈭� (x 鈭� 1) 鈰� 1}{(x + 3)^{2}} \\ = \frac{x + 3 鈭� x + 1}{(x + 3)^{2}} \\ = \frac{4}{(x + 3)^{2}} > 0 \end{aligned}$

Here the derivative is defined and continuous everywhere except at $x = - 3$ , so the function has no critical point such that $f^{'} (x) = 0$ , Therefore $f^{'} (x)$ is always greater than zero.

Step 3. Sign chart

Let us draw the sign chart,

Hence, the $f'$ always positive on $(- 鈭�, - 3) 鈭� (- 3, 鈭�)$ that is $f$ is always increasing $(- 鈭�, - 3) 鈭� (- 3, 鈭�)$ . The function is a concave up $(- 鈭�, - 3)$ and concave down at a point $(- 3, 鈭�)$ .