Find Where Increasing/Decreasing Using Derivatives f(x)=1/(x^2)

Problem

ƒ(x)=1/(x2)

Solution

1. Identify the domain of the function. The function ƒ(x)=1/(x2) is defined for all x except where the denominator is zero, which is x≠0

2. Find the derivative of the function. Rewrite the function as ƒ(x)=x(−2) and apply the power rule.

d(ƒ(x))/d(x)=−2*x(−3)

d(ƒ(x))/d(x)=−2/(x3)

3. Determine the critical points by setting the derivative equal to zero or finding where it is undefined. The derivative −2/(x3) is never zero. It is undefined at x=0 which is also where the original function is undefined.

4. Test the intervals created by the point x=0 The intervals to test are (−∞,0) and (0,∞)

5. Evaluate the sign of the derivative in the first interval (−∞,0) Choose a test point x=−1

ƒ′*(−1)=−2/((−1)3)=2

Since ƒ(x)′>0 the function is increasing on (−∞,0)

6. Evaluate the sign of the derivative in the second interval (0,∞) Choose a test point x=1

ƒ(1)′=−2/((1)3)=−2

Since ƒ(x)′<0 the function is decreasing on (0,∞)

Final Answer

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

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