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Implicit differentiation question

Descartesโ€™ folium is a curve defined by the equation ๐‘ฅ3 + ๐‘ฆ3 โˆ’ ๐‘Ž๐‘ฅ๐‘ฆ = 0. The case where ๐‘Ž = 9 is shown below.


๐‘‘๐‘ฅ
c) Hence determine the exact coordinates of the maximum point on the curve. [4]
d) Find the coordinates of the points on the curve for which there is no tangent defined [3]
Reply 1
Original post by lau321
Descartesโ€™ folium is a curve defined by the equation ๐‘ฅ3 + ๐‘ฆ3 โˆ’ ๐‘Ž๐‘ฅ๐‘ฆ = 0. The case where ๐‘Ž = 9 is shown below.


๐‘‘๐‘ฅ
c) Hence determine the exact coordinates of the maximum point on the curve. [4]
d) Find the coordinates of the points on the curve for which there is no tangent defined [3]
Can you upload the full question (as an image) and what youve done so far. The "hence" in part c) suggests it depends on an earlier part which youve not posted.
Do you have the image for the case where a=9?
Reply 3
Oh didn't notice that, sorry! It won't paste as an image, the earlier bit asked to find dy/dx which I believe is (3y2-x2)/(y2-3x). Sorry I can't get the full thing. So do I need to set dy/dx to 0 and then solve for x and y? Just confused on how to get to the answer, feel like I'm missing something obvious
Reply 4
Original post by lau321
Oh didn't notice that, sorry! It won't paste as an image, the earlier bit asked to find dy/dx which I believe is (3y2-x2)/(y2-3x). Sorry I can't get the full thing. So do I need to set dy/dx to 0 and then solve for x and y? Just confused on how to get to the answer, feel like I'm missing something obvious
for a gradient of 0 (max / min) your derivative will give you a relation between x and y which you can also plug into the original equation of the curve to get value(s) for x (and hence y) and you will need to sanity check that you are getting a max (or min) as required.

Tangent undefined basically means that the derivative must have 0 denominator, so proceed in a similar fashion.
Reply 5
Original post by lau321
Oh didn't notice that, sorry! It won't paste as an image, the earlier bit asked to find dy/dx which I believe is (3y2-x2)/(y2-3x). Sorry I can't get the full thing. So do I need to set dy/dx to 0 and then solve for x and y? Just confused on how to get to the answer, feel like I'm missing something obvious
Assuming a=9 (it would help to see the full question), Im getting a slightly different numerator. Otherwise as chavvo says, the gradient will be zero when the numerator is zero though the tangent being undefined would really just correspond to the crossing point. The line x=5 is a valid tangent (roughly for this question).

However, a tangent would be undefined at a crossing point as it doesnt touch the curve at that point (it must intersect assuming the curves have distinct gradients at that point). However, they may mean the derivative is undefined which is what is being suggested above.
(edited 1 month ago)
Reply 6
Original post by chavvo
for a gradient of 0 (max / min) your derivative will give you a relation between x and y which you can also plug into the original equation of the curve to get value(s) for x (and hence y) and you will need to sanity check that you are getting a max (or min) as required.

Tangent undefined basically means that the derivative must have 0 denominator, so proceed in a similar fashion.
perfect, thanks!

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