Find the Inflection Points y=-x^4+8x^3-8x+13

Problem

y=−x4+8*x3−8*x+13

Solution

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

d(y)/d(x)=−4*x3+24*x2−8

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

d2(y)/(d(x)2)=−12*x2+48*x

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

−12*x2+48*x=0

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

−12*x*(x−4)=0

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

x=0

x=4

6. Verify the concavity change by checking the sign of the second derivative on intervals around the points. Since the second derivative is a downward-opening parabola with roots at 0 and 4 it changes sign at both values.

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

y(0)=−(0)4+8*(0)3−8*(0)+13=13

y(4)=−(4)4+8*(4)3−8*(4)+13=−256+512−32+13=237

Final Answer

Inflection Points=(0,13),(4,237)

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