Find the Concavity f(x)=1/2x^4+2x^3

Problem

ƒ(x)=1/2*x4+2*x3

Solution

1. Find the first derivative by applying the power rule to each term of the function.

d(ƒ(x))/d(x)=2*x3+6*x2

2. Find the second derivative by differentiating the first derivative to determine the acceleration of the function.

d2(ƒ(x))/(d(x)2)=6*x2+12*x

3. Identify the critical points for concavity by setting the second derivative equal to zero and solving for x

6*x*(x+2)=0

x=0

x=−2

4. Test the intervals created by the critical points ((−∞,−2) (−2,0) and (0,∞) in the second derivative to determine the sign.
   For x=−3 6*(−3)2+12*(−3)=18>0 (Concave Up)
   For x=−1 6*(−1)2+12*(−1)=−6<0 (Concave Down)
   For x=1 6*(1)2+12*(1)=18>0 (Concave Up)

Final Answer

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

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