Calculate (dy)/(dx) of y + (17)/(y) = 2 x^(2) +12x and find all the points (x,y) on the curve with horizontal tangent lines. (not sure how to do this type of problem where they …

Question

Calculate
`(dy)/(dx)` of


`y + (17)/(y) = 2 x^(2) +12x`

and find all the points (x,y) on the curve with horizontal tangent lines.
(not sure how to do this type of problem where they don't give us points. Do we use the first derivative test?

Answer 1

`\bf y+\cfrac{17}{y}=2x62+12x\\\\ -----------------------------\\\\ \cfrac{dy}{dx}+17\left( -1y^(-2)(dy)/(dx) \right)=4x+12\implies \cfrac{dy}{dx}-\cfrac{17}{y^2}\cfrac{dy}{dx}=4x+12 \\\\\\ \cfrac{dy}{dx}\left(1-\cfrac{17}{y^2} \right)=4x+12\implies \cfrac{dy}{dx}=\cfrac{4x+12}{1-(17)/(y^2)}\\\\\\ \boxed{\cfrac{dy}{dx}=\cfrac{y^2(4x+12)}{y^2-17}}\\\\ -----------------------------\\\\ 0=\cfrac{y^2(4x+12)}{y^2-17}\implies 0=4x+12\implies \cfrac{-12}{4}=x\implies -3=x`