Find the Inflection Points (e^x)/(8+e^x)

Problem

ƒ(x)=(ex)/(8+ex)

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)′=((8+ex)*ex−e(ex)x)/((8+ex)2)

2. Simplify the first derivative by distributing and combining terms in the numerator.

ƒ(x)′=(8*ex+e(2*x)−e(2*x))/((8+ex)2)

ƒ(x)′=(8*ex)/((8+ex)2)

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

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

4. Factor and simplify the second derivative to find the points where ƒ(x)″=0

ƒ(x)″=(8*ex*(8+ex)*[(8+ex)−2*ex])/((8+ex)4)

ƒ(x)″=(8*ex*(8−ex))/((8+ex)3)

5. Solve for x by setting the numerator of the second derivative equal to zero.

8−ex=0

ex=8

x=ln(8)

6. Calculate the y-coordinate by substituting x=ln(8) back into the original function ƒ(x)

ƒ*(ln(8))=(eln(8))/(8+eln(8))

ƒ*(ln(8))=8/(8+8)

ƒ*(ln(8))=1/2

7. Verify the inflection point by checking for a sign change in ƒ(x)″ around x=ln(8) Since 8−ex changes from positive to negative, an inflection point exists.

Final Answer

Inflection Point=(ln(8),1/2)

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