Question

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

Answer 1

`\\ \tt\hookrightarrow f(x)=(x^2)/(x-6)`

Find inverse

`\\ \tt\hookrightarrow y=(x^2)/(x-6)`

`\\ \tt\hookrightarrow x=(x^2+6y)/(y)`

`\\ \tt\hookrightarrow f'(x)=(x^2+6x)/(x)=x+6`

Now

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

Find Horizontal tangency

`\\ \tt\hookrightarrow f(6)=((-6)^2)/(-6-6)`

`\\ \tt\hookrightarrow f(6)=(36)/(-12)=-3`

So

• (x,y) =(-6,-3)