Question

Please help: Determine the points at which the graph of the function has a horizontal tangent line.

Answer 1

$\\ \tt\hookrightarrow f(x)=\dfrac{x^2}{x-6}$

Find inverse

$\\ \tt\hookrightarrow y=\dfrac{x^2}{x-6}$

$\\ \tt\hookrightarrow x=\dfrac{x^2+6y}{y}$

$\\ \tt\hookrightarrow f'(x)=\dfrac{x^2+6x}{x}=x+6$

Now

$\\ \tt\hookrightarrow x+6=0\implies x=-6$

Find Horizontal tangency

$\\ \tt\hookrightarrow f(6)=\dfrac{(-6)^2}{-6-6}$

$\\ \tt\hookrightarrow f(6)=\dfrac{36}{-12}=-3$

So

• (x,y) =(-6,-3)