18. the derivative of a function f is f(x) = x/(x + 4) - (x + 3)/(x + 2). determine the interval(s) on which…

Explanation:

Step1: Find a common - denominator

First, find a common denominator for (f'(x)). The common denominator of (x + 4) and (x+2) is ((x + 4)(x + 2)). [f'(x)=\frac{x(x + 2)-(x + 3)(x + 4)}{(x + 4)(x + 2)}=\frac{x^{2}+2x-(x^{2}+4x+3x + 12)}{(x + 4)(x + 2)}=\frac{x^{2}+2x - x^{2}-7x - 12}{(x + 4)(x + 2)}=\frac{-5x - 12}{(x + 4)(x + 2)}]

Step2: Find the critical points

Set (f'(x)=0) and find where (f'(x)) is undefined. Set (-5x-12 = 0), then (x=-\frac{12}{5}=-2.4). (f'(x)) is undefined when (x=-4) and (x=-2).

Step3: Test the intervals

We have the intervals ((-\infty,-4)), ((-4,-2.4)), ((-2.4,-2)) and ((-2,\infty)). For (x=-5) (in the interval ((-\infty,-4))): (f'(-5)=\frac{-5\times(-5)-12}{(-5 + 4)(-5 + 2)}=\frac{25-12}{(-1)\times(-3)}=\frac{13}{3}>0). For (x = - 3) (in the interval ((-4,-2.4))): (f'(-3)=\frac{-5\times(-3)-12}{(-3 + 4)(-3 + 2)}=\frac{15 - 12}{1\times(-1)}=-3<0). For (x=-2.2) (in the interval ((-2.4,-2))): (f'(-2.2)=\frac{-5\times(-2.2)-12}{(-2.2 + 4)(-2.2 + 2)}=\frac{11-12}{1.8\times(-0.2)}=\frac{-1}{-0.36}>0). For (x=0) (in the interval ((-2,\infty))): (f'(0)=\frac{-5\times0-12}{(0 + 4)(0 + 2)}=\frac{-12}{8}<0).

Answer:

The function (f) is increasing on the intervals ((-\infty,-4)) and ((-2.4,-2)).