`f(x) = (x^2)/(x^2 + 1)` Determine the point(s) at which the graph of the function has a horizontal tangent line.

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Chapter 2, 2.3 - Problem 74 - Calculus of a Single Variable (10th Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Given: `f(x)=x^2/(x^2+1)`

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Find the derivative of the function using the Quotient Rule. Set the derivative equal zero to find the critical x value(s).

`f'(x)=[(x^2+1)(2x)-(x^2)(2x)]/(x^2+1)^2`

`f'(x)=(2x^3+2x-2x^3)/(x^2+1)^2`

`f'(x)=(2x)/(x^2+1)^2=0`

`2x=0`

`x=0`

Plug in the critical x values into the f(x) equation. 

`f(x)=x^2/(x^2+1)`

`f(0)=0/1=0`

The graph of the function will have a horizontal tangent line at the

point (0, 0). 

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