Find the Concavity f(x)=x^(1/3)(x+4)

Problem

ƒ(x)=x^(1/3)*(x+4)

Solution

1. Distribute the term x^(1/3) to simplify the function before differentiation.

ƒ(x)=x^(4/3)+4*x^(1/3)

2. Find the first derivative ƒ(x)^′ using the power rule.

ƒ(x)^′=4/3*x^(1/3)+4/3*x^(−2/3)

3. Find the second derivative ƒ(x)^″ by differentiating ƒ(x)^′

ƒ(x)^″=4/9*x^(−2/3)−8/9*x^(−5/3)

4. Factor the second derivative to find critical points for concavity.

ƒ(x)^″=4/9*x^(−5/3)*(x−2)

5. Identify critical values where ƒ(x)^″=0 or ƒ(x)^″ is undefined.

x=2

x=0

6. Test intervals (−∞,0) (0,2) and (2,∞) in ƒ(x)^″ to determine the sign.
   For x=−1 ƒ^″*(−1)=4/9*(−1)^(−5/3)*(−1−2)=4/9*(−1)*(−3)=4/3>0 (Concave Up)
   For x=1 ƒ(1)^″=4/9*(1)^(−5/3)*(1−2)=4/9*(1)*(−1)=−4/9<0 (Concave Down)
   For x=3 ƒ(3)^″=4/9*(3)^(−5/3)*(3−2)=4/9*(3)^(−5/3)*(1)>0 (Concave Up)

Final Answer

Concave Up: *(−∞,0)∪(2,∞), Concave Down: *(0,2)

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