Find the Inflection Points f(x)=4x^3-x^4

Problem

ƒ(x)=4*x3−x4

Solution

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

d(ƒ(x))/d(x)=12*x2−4*x3

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

d2(ƒ(x))/(d(x)2)=24*x−12*x2

3. Set the second derivative to zero to find potential inflection points where the concavity might change.

24*x−12*x2=0

4. Factor the equation to solve for the critical values of x

12*x*(2−x)=0

5. Solve for x by setting each factor to zero.

x=0

x=2

6. Verify the concavity change by testing intervals around the critical values in the second derivative. For x<0 ƒ(x)″<0 For 0<x<2 ƒ(x)″>0 For x>2 ƒ(x)″<0 Since the sign changes at both points, both are inflection points.

7. Calculate the y-coordinates by substituting the x values back into the original function ƒ(x)

ƒ(0)=4*(0)3−(0)4=0

ƒ(2)=4*(2)3−(2)4=32−16=16

Final Answer

Inflection Points: *(0,0),(2,16)

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