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

Problem

ƒ(x)=x3−3*x+1

Solution

1. Find the derivative of the function to determine the slope of the tangent line at any point x

d(ƒ(x))/d(x)=3*x2−3

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

3*x2−3=0

3*(x2−1)=0

x2=1

x=1,x=−1

3. Test intervals created by the critical points (−∞,−1) (−1,1) and (1,∞) to determine the sign of ƒ(x)′
   For x=−2 ƒ′*(−2)=3*(−2)2−3=9>0 (Increasing)
   For x=0 ƒ(0)′=3*(0)2−3=−3<0 (Decreasing)
   For x=2 ƒ(2)′=3*(2)2−3=9>0 (Increasing)

4. Conclude the intervals based on where the derivative is positive (increasing) or negative (decreasing).

Final Answer

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

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