Find the Concavity y=-x^4+4x^3-4x+2

Problem

y=−x^4+4*x^3−4*x+2

Solution

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

d(y)/d(x)=−4*x^3+12*x^2−4

2. Find the second derivative by differentiating the first derivative with respect to x

d^2(y)/(d(x)^2)=−12*x^2+24*x

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

−12*x^2+24*x=0

−12*x*(x−2)=0

x=0,x=2

4. Test the intervals created by the critical points ((−∞,0) (0,2) and (2,∞) in the second derivative to determine the sign.
   For x=−1 −12*(−1)^2+24*(−1)=−36 (Negative)
   For x=1 −12*(1)^2+24*(1)=12 (Positive)
   For x=3 −12*(3)^2+24*(3)=−36 (Negative)

5. Identify the concavity based on the signs: concave down where the second derivative is negative and concave up where it is positive.
   Concave Down: (−∞,0)∪(2,∞)
   Concave Up: (0,2)

Final Answer

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

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