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 ƒ(x) 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 x

6*(x2−x−2)=0

6*(x−2)*(x+1)=0

4. Identify the critical values by solving the factored equation.

x=2,x=−1

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

6. Determine the intervals based on the signs of the derivative.

   ƒ(x)$i*s(i)*n*c*r*e*a*s(i)*n*g*o*n$(−∞,−1)∪(2,∞)

   ƒ(x)$i*s(d(e))*c*r*e*a*s(i)*n*g*o*n$(−1,2)

Final Answer

Increasing: *(−∞,−1)∪(2,∞), Decreasing: *(−1,2)

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