Find the Concavity f(x)=3x(x-2)^3

Problem

ƒ(x)=3*x*(x−2)3

Solution

1. Find the first derivative using the product rule and the chain rule.

ƒ(x)′=3*(x−2)3+3*x⋅3*(x−2)2

ƒ(x)′=3*(x−2)3+9*x*(x−2)2

2. Simplify the first derivative by factoring out the common term 3*(x−2)2

ƒ(x)′=3*(x−2)2*((x−2)+3*x)

ƒ(x)′=3*(x−2)2*(4*x−2)

ƒ(x)′=6*(x−2)2*(2*x−1)

3. Find the second derivative using the product rule on the simplified first derivative.

ƒ(x)″=6⋅2*(x−2)*(1)⋅(2*x−1)+6*(x−2)2⋅2

ƒ(x)″=12*(x−2)*(2*x−1)+12*(x−2)2

4. Simplify the second derivative by factoring out 12*(x−2)

ƒ(x)″=12*(x−2)*((2*x−1)+(x−2))

ƒ(x)″=12*(x−2)*(3*x−3)

ƒ(x)″=36*(x−2)*(x−1)

5. Identify the critical points for concavity by setting ƒ(x)″=0

36*(x−2)*(x−1)=0

x=1,x=2

6. Test the intervals (−∞,1) (1,2) and (2,∞) in the second derivative to determine the sign.

For *x<1,ƒ(0)″=36*(−2)*(−1)=72>0⇒Concave Up

For *1<x<2,ƒ(1.5)″=36*(−0.5)*(0.5)=−9<0⇒Concave Down

For *x>2,ƒ(3)″=36*(1)*(2)=72>0⇒Concave Up

Final Answer

Concave Up on *(−∞,1)∪(2,∞), Concave Down on *(1,2)

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