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

Problem

ƒ(x)=x4−2*x2

Solution

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

d(ƒ(x))/d(x)=4*x3−4*x

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

d2(ƒ(x))/(d(x)2)=12*x2−4

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

12*x2−4=0

4. Solve for x by isolating the variable and taking the square root of both sides.

12*x2=4

x2=1/3

x=±1/√(,3)

5. Rationalize the denominator to express the xcoordinates in standard form.

x=±√(,3)/3

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

ƒ(√(,3)/3)=(√(,3)/3)4−2*(√(,3)/3)2

ƒ(√(,3)/3)=9/81−2*(3/9)

ƒ(√(,3)/3)=1/9−6/9=−5/9

7. Verify the concavity change by checking the sign of ƒ(x)″ around the points; since ƒ(x)″ is a parabola opening upward, it changes sign at both roots.

Final Answer

Inflection Points=(−√(,3)/3,−5/9),(√(,3)/3,−5/9)

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