Find Where Increasing/Decreasing Using Derivatives f(x)=x^4-4x^3+10

Problem

ƒ(x)=x4−4*x3+10

Solution

1. Find the derivative of the function ƒ(x) using the power rule to determine the slope of the tangent line.

d(ƒ(x))/d(x)=4*x3−12*x2

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

4*x3−12*x2=0

4*x2*(x−3)=0

x=0,x=3

3. Test intervals created by the critical points (−∞,0) (0,3) and (3,∞) by plugging a test value from each interval into ƒ(x)′
   For (−∞,0) test x=−1 4*(−1)3−12*(−1)2=−16<0 (Decreasing)
   For (0,3) test x=1 4*(1)3−12*(1)2=−8<0 (Decreasing)
   For (3,∞) test x=4 4*(4)3−12*(4)2=64>0 (Increasing)

4. Determine the intervals where the function is increasing (where ƒ(x)′>0 and decreasing (where ƒ(x)′<0.

Final Answer

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

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