Find the Concavity f(x)=x^4-6x^3+2x+8

Problem

ƒ(x)=x4−6*x3+2*x+8

Solution

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

d(ƒ(x))/d(x)=4*x3−18*x2+2

2. Find the second derivative by differentiating the first derivative to determine the concavity.

d2(ƒ(x))/(d(x)2)=12*x2−36*x

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

12*x*(x−3)=0

x=0

x=3

4. Test intervals created by the zeros (x<0 0<x<3 and x>3 in the second derivative to determine the sign.

For *x=−1:12*(−1)2−36*(−1)=48>0

For *x=1:12*(1)2−36*(1)=−24<0

For *x=4:12*(4)2−36*(4)=48>0

5. Determine concavity based on the sign of the second derivative: positive indicates concave up, and negative indicates concave down.

Concave up on *(−∞,0)∪(3,∞)

Concave down on *(0,3)

Final Answer

ƒ(x)* is concave up on *(−∞,0)∪(3,∞)* and concave down on *(0,3)

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