Find Where Increasing/Decreasing Using Derivatives f(x)=-12x^5+135x^4-400x^3

Problem

ƒ(x)=−12*x5+135*x4−400*x3

Solution

1. Find the derivative of the function using the power rule to determine the rate of change.

d(ƒ(x))/d(x)=−60*x4+540*x3−1200*x2

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

−60*x2*(x2−9*x+20)=0

3. Factor the quadratic expression inside the parentheses to find all roots.

−60*x2*(x−4)*(x−5)=0

4. Solve for x to find the critical values where the slope of the function is zero.

x=0,x=4,x=5

5. Test the intervals created by the critical points (−∞,0) (0,4) (4,5) and (5,∞) in the derivative ƒ(x)′ to determine the sign.
   For x=−1 ƒ′*(−1)=−60*(−1)2*(−1−4)*(−1−5)=−1800 (Decreasing)
   For x=1 ƒ(1)′=−60*(1)2*(1−4)*(1−5)=−720 (Decreasing)
   For x=4.5 ƒ(4.5)′=−60*(4.5)2*(4.5−4)*(4.5−5)=303.75 (Increasing)
   For x=6 ƒ(6)′=−60*(6)2*(6−4)*(6−5)=−4320 (Decreasing)

6. Determine the intervals based on the sign of the derivative. The function increases where ƒ(x)′>0 and decreases where ƒ(x)′<0

Final Answer

Increasing: *(4,5), Decreasing: *(−∞,0)∪(0,4)∪(5,∞)

Want more problems? Check here!