Find the Inflection Points y=-x^4+4x^3-4x+2

Problem

y=−x4+4*x3−4*x+2

Solution

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

d(y)/d(x)=−4*x3+12*x2−4

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

d2(y)/(d(x)2)=−12*x2+24*x

3. Set the second derivative to zero to find potential inflection points.

−12*x2+24*x=0

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

−12*x*(x−2)=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. Since d2(y)/(d(x)2) is a downward-opening parabola with roots at 0 and 2 the sign changes from negative to positive at x=0 and from positive to negative at x=2

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

y(0)=−(0)4+4*(0)3−4*(0)+2=2

y(2)=−(2)4+4*(2)3−4*(2)+2=−16+32−8+2=10

Final Answer

Inflection Points: *(0,2),(2,10)

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