Find Where Increasing/Decreasing Using Derivatives f(x)=x^3+3x^2-24x+6

Problem

ƒ(x)=x3+3*x2−24*x+6

Solution

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

d(ƒ(x))/d(x)=3*x2+6*x−24

2. Set the derivative to zero to find the critical points where the slope of the function is horizontal.

3*x2+6*x−24=0

3. Factor the quadratic equation to solve for x

3*(x2+2*x−8)=0

3*(x+4)*(x−2)=0

4. Identify the critical values from the factors.

x=−4

x=2

5. Test the intervals created by the critical values ((−∞,−4) (−4,2) and (2,∞) in the derivative ƒ(x)′ to determine the sign.
   For x=−5 ƒ′*(−5)=3*(−5)2+6*(−5)−24=21>0 (Increasing)
   For x=0 ƒ(0)′=3*(0)2+6*(0)−24=−24<0 (Decreasing)
   For x=3 ƒ(3)′=3*(3)2+6*(3)−24=21>0 (Increasing)

Final Answer

Increasing: *(−∞,−4)∪(2,∞), Decreasing: *(−4,2)

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