Find Where Increasing/Decreasing Using Derivatives f(x)=2x^3+3x^2-12x

Problem

ƒ(x)=2*x3+3*x2−12*x

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)=6*x2+6*x−12

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

6*x2+6*x−12=0

3. Factor the quadratic equation to solve for the critical values of x

6*(x2+x−2)=0

6*(x+2)*(x−1)=0

x=−2,x=1

4. Test the intervals created by the critical points (−∞,−2) (−2,1) and (1,∞) by plugging values into ƒ(x)′
   For x=−3 ƒ′*(−3)=6*(−3+2)*(−3−1)=24>0 (Increasing)
   For x=0 ƒ(0)′=6*(0+2)*(0−1)=−12<0 (Decreasing)
   For x=2 ƒ(2)′=6*(2+2)*(2−1)=24>0 (Increasing)

5. Identify 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)∪(1,∞), Decreasing: *(−2,1)

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