Find the Concavity f(x)=2x^4-12x^2

Problem

ƒ(x)=2*x4−12*x2

Solution

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

d(ƒ(x))/d(x)=8*x3−24*x

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

d2(ƒ(x))/(d(x)2)=24*x2−24

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

24*x2−24=0

24*(x2−1)=0

x=1,x=−1

4. Test intervals created by the points x=−1 and x=1 to determine the sign of the second derivative.
   For (−∞,−1) test x=−2 24*(−2)2−24=72>0 (Concave Up).
   For (−1,1) test x=0 24*(0)2−24=−24<0 (Concave Down).
   For (1,∞) test x=2 24*(2)2−24=72>0 (Concave Up).

Final Answer

Concave Up: *(−∞,−1)∪(1,∞), Concave Down: *(−1,1)

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