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

Problem

ƒ(x)=(x2)/(x2+4)

Solution

1. Find the first derivative using the quotient rule d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

ƒ(x)′=((x2+4)*(2*x)−(x2)*(2*x))/((x2+4)2)

ƒ(x)′=(2*x3+8*x−2*x3)/((x2+4)2)

ƒ(x)′=(8*x)/((x2+4)2)

2. Find the second derivative by applying the quotient rule again to ƒ(x)′

ƒ(x)″=((x2+4)2*(8)−(8*x)*(2*(x2+4)*(2*x)))/((x2+4)4)

3. Simplify the second derivative by factoring out (x2+4) from the numerator.

ƒ(x)″=(8*(x2+4)−32*x2)/((x2+4)3)

ƒ(x)″=(8*x2+32−32*x2)/((x2+4)3)

ƒ(x)″=(32−24*x2)/((x2+4)3)

4. Identify potential inflection points by setting ƒ(x)″=0

32−24*x2=0

24*x2=32

x2=4/3

x=±2/√(,3)

5. Rationalize the x-coordinates to get standard form.

x=±(2√(,3))/3

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

ƒ*(±2/√(,3))=4/3/(4/3+4)

ƒ*(±2/√(,3))=4/3/16/3

ƒ*(±2/√(,3))=1/4

Final Answer

Inflection Points: *((2√(,3))/3,1/4),(−(2√(,3))/3,1/4)

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