$\text { Determine all values of } x \text {, (if any), at which the graph of the function has a }$ horizontal tangent.
$y ( x ) = \frac { 9 } { x - 6 }$
A) $x = 9$ and $x = - 6$
B) $x = 9$
C) $x = 9$ and $x = 6$
D) $x = 6$
E) The graph has no horizontal tangents.

Question

Quizplus · Accepted Answer

A horizontal tangent occurs where the derivative of the function is zero. The derivative of $y(x) = \frac{9}{x - 6}$ is $y'(x) = -\frac{9}{(x - 6)^2}$. Since the denominator is squared, it is always positive (except when $x = 6$, where the function is undefined), and the numerator is a constant, so the derivative can never be zero. Therefore, the graph of the function has no horizontal tangents.

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