How do you find the points where the graph of the function $F (x) = \frac{x}{x^{2} + 4}$ $F(x)=x/(x^2+4)$ has horizontal tangents and what is the equation?

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How do you find the points where the graph of the function $F (x) = \frac{x}{x^{2} + 4}$ $F(x)=x/(x^2+4)$ has horizontal tangents and what is the equation?

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Answer 1 · 2016-03-28T05:25:15

The tangent is horizontal at $(2, \frac{1}{4})$ $(2, 1/4)$ and $(- 2, - \frac{1}{4})$ $(-2, -1/4)$ .
The equations of these tangents parallel to x-axis are $y = \pm \frac{1}{4}$ $y=+-1/4$ .

Explanation:

y = $\frac{x}{x^{2} + 4}$ $x/(x^2+4)$
$y' = (4-x^2)/(x^2+4)^2$ = 0, when x = $+-2$ .
The points at which the tangents are horizontal are $(2, 1/4)$ and $(-2,-1/4).$
The equations of 0-slope horizontal tangents are y= $+-1/4.$

The graph passes through the origin, with y increasing between the turning points $(-2, -1/4)$ and $(2, 1/4)$ and decreasing to the limit 0, outside $-2<=x<2$ , as $xto+-oo$ ..

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How do you find the points where the graph of the function $F (x) = \frac{x}{x^{2} + 4}$ $F(x)=x/(x^2+4)$ has horizontal tangents and what is the equation?

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How do you find the points where the graph of the function $F (x) = \frac{x}{x^{2} + 4}$ $F(x)=x/(x^2+4)$ has horizontal tangents and what is the equation?