Find the Concavity f(x)=x^4-6x^2

Problem

ƒ(x)=x4−6*x2

Solution

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

d(ƒ(x))/d(x)=4*x3−12*x

2. Find the second derivative by differentiating the first derivative to determine the rate of change of the slope.

d2(ƒ(x))/(d(x)2)=12*x2−12

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

12*x2−12=0

12*(x2−1)=0

x=±1

4. Test intervals to determine the sign of the second derivative on (−∞,−1) (−1,1) and (1,∞)
   For x=−2 12*(−2)2−12=36>0 (Concave Up)
   For x=0 12*(0)2−12=−12<0 (Concave Down)
   For x=2 12*(2)2−12=36>0 (Concave Up)

Final Answer

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

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