At which point(s) does the graph of the function&nbsp;f(x)=x2x-1&nbsp;have a horizontal tangent line?

Question

At which point(s) does the graph of the function&amp;nbsp;f(x)=x2x-1&amp;nbsp;have a horizontal tangent line?

orangepaperiz7 · Accepted Answer

The slope of the tangent line of a graph y=f(x) at a point

x_{0}

is given by the derivative of f at that point, that is,

f^{'} (x_{0})

A horizontal tangent line implies a slope of 0, so our goal is to find the points at
which the derivative f(x) evaluates to 0.
Using the quotient rule, we find the derivative as

f^{'} (x) = \frac{d}{d x} \frac{x^{2}}{x - 1}

= \frac{(x - 1) (\frac{d}{d x} x^{2}) - x^{2} (\frac{d}{d x} (x - 1))}{{(x - 1)}^{2}}

= \frac{2 x (x - 1) - x^{2} (1)}{{(x - 1)}^{2}}

= \frac{2 x^{2} - 2 x - x^{2}}{{(x - 1)}^{2}}

= \frac{x^{2} - 2 x}{{(x - 1)}^{2}}

= \frac{x (x - 2)}{{(x - 1)}^{2}}

Setting this equal to zero, we get

\frac{x (x - 2)}{{(x - 1)}^{2}} = 0

\Rightarrow x (x - 2) = 0

\Rightarrow x = 0

or x=2
Thus, the graph of f(x) has a horizontal tangent line at x=0 and x=2, that is, at the points
(0,0),(2,4)

At which point(s) does the graph of the

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