Find Where Increasing/Decreasing Using Derivatives f(x)=3x^4-4x^3

Problem

ƒ(x)=3*x4−4*x3

Solution

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

d(ƒ(x))/d(x)=12*x3−12*x2

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

12*x2*(x−1)=0

x=0

x=1

3. Test intervals created by the critical points (−∞,0) (0,1) and (1,∞) by plugging values into ƒ(x)′ to check the sign.
   For (−∞,0) test x=−1 12*(−1)2*(−1−1)=−24 (Negative)
   For (0,1) test x=0.5 12*(0.5)2*(0.5−1)=−1.5 (Negative)
   For (1,∞) test x=2 12*(2)2*(2−1)=48 (Positive)

4. Determine behavior based on the signs: the function is decreasing where ƒ(x)′<0 and increasing where ƒ(x)′>0

Final Answer

Increasing: *(1,∞), Decreasing: *(−∞,0)∪(0,1)

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