Find the Inflection Points f(x)=x^4-24x^2+12x

Problem

ƒ(x)=x4−24*x2+12*x

Solution

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

d(ƒ(x))/d(x)=4*x3−48*x+12

2. Find the second derivative by differentiating the first derivative.

d2(ƒ(x))/(d(x)2)=12*x2−48

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

12*x2−48=0

4. Solve for x by isolating the variable.

12*x2=48

x2=4

x=±2

5. Verify the concavity change by testing intervals around x=−2 and x=2 in the second derivative. Since ƒ(x)″ is a parabola opening upward, it is positive for x<−2 negative for −2<x<2 and positive for x>2

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

ƒ(2)=(2)4−24*(2)2+12*(2)=16−96+24=−56

ƒ*(−2)=(−2)4−24*(−2)2+12*(−2)=16−96−24=−104

Final Answer

Inflection Points=(2,−56),(−2,−104)

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