Find the Inflection Points f(x)=x^4-2x^2+3

Problem

ƒ(x)=x4−2*x2+3

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 potential inflection points where the concavity might change.

12*x2−4=0

4. Solve for x by isolating the variable.

12*x2=4

x2=1/3

x=±1/√(,3)

5. Rationalize the denominator for the xvalues.

x=±√(,3)/3

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

ƒ(√(,3)/3)=(1/3)2−2*(1/3)+3

ƒ(√(,3)/3)=1/9−6/9+27/9

ƒ(√(,3)/3)=22/9

ƒ*(−√(,3)/3)=22/9

7. Verify the concavity change by checking the sign of ƒ(x)″ around the points. Since ƒ(x)″ is a parabola opening upward with roots at these values, the sign changes from positive to negative and back to positive, confirming they are inflection points.

Final Answer

Inflection Points=(√(,3)/3,22/9),(−√(,3)/3,22/9)

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