Help with solving graph equation (use of implicit differentiation)

Equation of graph : 2x^{5 }+ 2y^{2}= 5xy

The vertical tangent line and the horizontal tangent line to the curve intersect at the point A(a,b). Find the coordinates of A.

The graph is shown on my paper. Maybe you guys can use an online graphing app to see how it looks like or use scientific calculator.

Anyway, please help!

Re: Help with solving graph equation (use of implicit differentiation)

When you implicitly differentiate with respect to $\displaystyle x$, what do you get for $\displaystyle \frac{dy}{dx}$ ?

Re: Help with solving graph equation (use of implicit differentiation)

Re: Help with solving graph equation (use of implicit differentiation)

Yes, good work! I would write this as:

$\displaystyle \frac{dy}{dx}=\frac{5(y-2x^4)}{4y-5x}$

Now, equating the numerator to zero will tell us where the curve has a horizontal tangent, white equating the denominator to zero will tell us where the vertical tangent occurs. What do you find?

Re: Help with solving graph equation (use of implicit differentiation)

Horizontal tangent:

y-2x^4 = 0

y = 2x^4

Vertical tangent:

4y-5x = 0

Do i substitute at this point?

Re: Help with solving graph equation (use of implicit differentiation)

Yes, exactly! :D Careful though...you will get a meaningless point that you should discard.