Find the Inflection Points x^4-2x^2

Problem

x4−2*x2

Solution

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

(d(x4)−2*x2)/d(x)=4*x3−4*x

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

(d(4)*x3−4*x)/d(x)=12*x2−4

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

12*x2−4=0

4. Solve for x by isolating the variable.

12*x2=4

x2=4/12

x2=1/3

x=±√(,1/3)

5. Simplify the coordinates by rationalizing the denominator.

x=±√(,3)/3

6. Verify the concavity change by checking the sign of the second derivative on intervals around the points. Since the second derivative is a parabola opening upward, it changes sign at both roots.

Final Answer

x=√(,3)/3,−√(,3)/3

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