Find Where Increasing/Decreasing Using Derivatives f(x)=1/4x^4-2x^2

Problem

ƒ(x)=1/4*x4−2*x2

Solution

1. Find the derivative of the function using the power rule to determine the rate of change.

d(ƒ(x))/d(x)=x3−4*x

2. Identify critical points by setting the derivative equal to zero and solving for x

x3−4*x=0

x*(x2−4)=0

x*(x−2)*(x+2)=0

x=−2,0,2

3. Test intervals created by the critical points (−∞,−2) (−2,0) (0,2) and (2,∞) in the derivative ƒ(x)′ to determine the sign.
   For x=−3 ƒ′*(−3)=(−3)3−4*(−3)=−15 (Negative)
   For x=−1 ƒ′*(−1)=(−1)3−4*(−1)=3 (Positive)
   For x=1 ƒ(1)′=(1)3−4*(1)=−3 (Negative)
   For x=3 ƒ(3)′=(3)3−4*(3)=15 (Positive)

4. Determine behavior based on the sign of the derivative: the function increases where ƒ(x)′>0 and decreases where ƒ(x)′<0
   Increasing: (−2,0)∪(2,∞)
   Decreasing: (−∞,−2)∪(0,2)

Final Answer

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

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