Find the Inflection Points f(x)=x^4-4x^3+10

Problem

ƒ(x)=x4−4*x3+10

Solution

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

d(ƒ(x))/d(x)=4*x3−12*x2

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

d2(ƒ(x))/(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 expression to solve for the critical values of x

12*x*(x−2)=0

5. Solve for x by setting each factor equal 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 (concave up). For 0<x<2 ƒ(x)″<0 (concave down). For x>2 ƒ(x)″>0 (concave up). Since the concavity changes at both points, they are inflection points.

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

ƒ(0)=(0)4−4*(0)3+10=10

ƒ(2)=(2)4−4*(2)3+10=16−32+10=−6

Final Answer

Inflection Points: *(0,10),(2,−6)

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