Find Where Increasing/Decreasing Using Derivatives f(x)=x^3+6x^2+9x

Problem

ƒ(x)=x^3+6*x^2+9*x

Solution

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

d(ƒ(x))/d(x)=3*x^2+12*x+9

2. Set the derivative to zero to find the critical points where the function might change direction.

3*x^2+12*x+9=0

3. Factor the quadratic equation to solve for x

3*(x^2+4*x+3)=0

3*(x+3)*(x+1)=0

4. Identify the critical values by solving for x in each factor.

x=−3

x=−1

5. Test intervals created by the critical values ((−∞,−3) (−3,−1) and (−1,∞) by plugging a test point from each into ƒ(x)^′ to check the sign.
   For (−∞,−3) test x=−4 ƒ^′*(−4)=3*(−4+3)*(−4+1)=9>0 (Increasing).
   For (−3,−1) test x=−2 ƒ^′*(−2)=3*(−2+3)*(−2+1)=−3<0 (Decreasing).
   For (−1,∞) test x=0 ƒ(0)^′=3*(0+3)*(0+1)=9>0 (Increasing).

Final Answer

Increasing: *(−∞,−3)∪(−1,∞), Decreasing: *(−3,−1)

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