Use implicit differentiation to find the largest y-value in the loop of the Folium of Descartes, which is given by :$ x^3 + y^3 - 3xy = 0$. My professor didn't even explain how to do this. I asked the TA,and they told me to equal the first derivative to $0$ and solve for $x$ and then see if the second derivative is less than $0$? Can someone explain this problem to me?
First, we do the implicit derivative to simplify our equation. Because maxima/minima occur when $f'(x)=0$, we take the implicit derivative and set it equal to 0. We should end up with a few maxima/minima. To tell whether if it is the maxima or minima, we use the second derivative test to see whether it is a maxima or minima(If $f''(x)&lt;0$ then it is a maxima ).
Yep, that's exactly what you gotta do: ' equal the first derivative to 0 and solve for x and then see if the second derivative is less than 0'... Assuming you know implicit differentiation it should be easy - and in case you don't - you do know it, it's basically the chain rule, differentiate everything by x, applying chain rule to y.