Find Where Increasing/Decreasing Using Derivatives f(x)=x^3+18x^2+2

Problem

ƒ(x)=x3+18*x2+2

Solution

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

d(ƒ(x))/d(x)=3*x2+36*x

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

3*x2+36*x=0

3*x*(x+12)=0

x=0

x=−12

3. Test the intervals created by the critical points (−∞,−12) (−12,0) and (0,∞) in the derivative ƒ(x)′ to determine the sign.
   For x=−13 3*(−13)2+36*(−13)=507−468=39>0 (Increasing)
   For x=−1 3*(−1)2+36*(−1)=3−36=−33<0 (Decreasing)
   For x=1 3*(1)2+36*(1)=3+36=39>0 (Increasing)

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

Final Answer

Increasing on: *(−∞,−12)∪(0,∞)

Decreasing on: *(−12,0)

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