Find the Inflection Points f(x)=6x^4-36x^2

Problem

ƒ(x)=6*x4−36*x2

Solution

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

d(ƒ(x))/d(x)=24*x3−72*x

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

d2(ƒ(x))/(d(x)2)=72*x2−72

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

72*x2−72=0

4. Solve for x by factoring the quadratic equation.

72*(x2−1)=0

72*(x−1)*(x+1)=0

x=1

x=−1

5. Verify the concavity change by testing intervals around the critical values in the second derivative. Since ƒ(x)″ is a parabola opening upward, it is negative between −1 and 1 and positive elsewhere, confirming both are inflection points.

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

ƒ(1)=6*(1)4−36*(1)2=6−36=−30

ƒ*(−1)=6*(−1)4−36*(−1)2=6−36=−30

Final Answer

Inflection Points: *(1,−30),(−1,−30)

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