Find Where Increasing/Decreasing Using Derivatives f(x)=(x^2)/(x^2-4)

Problem

ƒ(x)=(x2)/(x2−4)

Solution

1. Find the domain of the function by identifying where the denominator is zero.

x2−4=0

x=±2

The domain is (−∞,−2)∪(−2,2)∪(2,∞)

2. Calculate the derivative using the quotient rule d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

d(ƒ(x))/d(x)=((x2−4)*(2*x)−(x2)*(2*x))/((x2−4)2)

3. Simplify the numerator to find the expression for ƒ(x)′

d(ƒ(x))/d(x)=(2*x3−8*x−2*x3)/((x2−4)2)

d(ƒ(x))/d(x)=(−8*x)/((x2−4)2)

4. Identify critical numbers and points of discontinuity by setting the derivative to zero and checking where it is undefined.

−8*x=0⇒x=0

The derivative is undefined at x=−2 and x=2 which are the vertical asymptotes.

5. Test intervals created by the critical point x=0 and the discontinuities x=−2,2 to determine the sign of ƒ(x)′
   For (−∞,−2) test x=−3 ƒ′*(−3)=24/25>0 (Increasing).
   For (−2,0) test x=−1 ƒ′*(−1)=8/9>0 (Increasing).
   For (0,2) test x=1 ƒ(1)′=(−8)/9<0 (Decreasing).
   For (2,∞) test x=3 ƒ(3)′=(−24)/25<0 (Decreasing).

Final Answer

Increasing: *(−∞,−2)∪(−2,0), Decreasing: *(0,2)∪(2,∞)

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