Find Where Increasing/Decreasing Using Derivatives f(x)=x^3-12x+1

Problem

ƒ(x)=x3−12*x+1

Solution

1. Find the derivative of the function to determine the rate of change.

d(ƒ(x))/d(x)=3*x2−12

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

3*x2−12=0

3*x2=12

x2=4

x=±2

3. Test intervals created by the critical points x=−2 and x=2 to determine the sign of ƒ(x)′
   For (−∞,−2) test x=−3 3*(−3)2−12=15>0 (Increasing).
   For (−2,2) test x=0 3*(0)2−12=−12<0 (Decreasing).
   For (2,∞) test x=3 3*(3)2−12=15>0 (Increasing).

4. Conclude the intervals based on the sign of the derivative.
   The function is increasing where ƒ(x)′>0 and decreasing where ƒ(x)′<0

Final Answer

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

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