Find Where Increasing/Decreasing Using Derivatives f(x)=x^4-6x^2

Problem

ƒ(x)=x4−6*x2

Solution

1. Find the derivative of the function using the power rule to determine the slope of the tangent line.

d(ƒ(x))/d(x)=4*x3−12*x

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

4*x3−12*x=0

4*x*(x2−3)=0

x=0,x=√(,3),x=−√(,3)

3. Test intervals created by the critical points (−∞,−√(,3)) (−√(,3),0) (0,√(,3)) and (√(,3),∞) by plugging values into ƒ(x)′

ƒ′*(−2)=4*(−2)3−12*(−2)=−8<0

ƒ′*(−1)=4*(−1)3−12*(−1)=8>0

ƒ(1)′=4*(1)3−12*(1)=−8<0

ƒ(2)′=4*(2)3−12*(2)=8>0

4. Determine intervals of increase where ƒ(x)′>0 and decrease where ƒ(x)′<0

Increasing: *(−√(,3),0)∪(√(,3),∞)

Decreasing: *(−∞,−√(,3))∪(0,√(,3))

Final Answer

Increasing: *(−√(,3),0)∪(√(,3),∞), Decreasing: *(−∞,−√(,3))∪(0,√(,3))

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