Can someone explain this answer "in layman's terms"?

[iwantcalculus]

The question is :

Show that the curve defined implicitly by the equation

X*Y^3 + x^3*Y = 4

has no horizontal tangent.The answer is :

Can someone please explain what steps are used in order to obtain the answer in a simple way for a calculus beginner? This answer is not helping me understand the question and how to answer the question ( It is too complicated and gets me confused).

And if you have similar questions, please show me them so I can practice on them after I understand this question in order to be comfortable with this type of questions.

 

[MisterX]

A curve is given $f(x,y) = c$. To see if there are any horizontal tangents, you need to get $y' = \frac{dy}{dx}$. So then you take the derivative of both sides of the equation $f(x,y) = c$ resulting in some other true equation $f'(x,y) = 0$. You can obtain $f'$ by implicit differentiation. Now somewhere in $f'$ there will be a $dy/dx$ aka $y'$. You need to extract that somehow and solve for it. How you do this will depend on the the function you were given. In this case $y'$ is that that ratio of two polynomials.

To find if the function has a horizontal tangent we solve $dy/dx = 0$. In our case $dy/dx$ is a fraction of two things $$y' = \frac{a(x,y)}{b(x,y)} = 0$$ can only be true if $a(x,y)= 0$. So to find solutions set the numerator to zero (note however that $a(x,y) = 0$ and $b(x,y) = 0$ is a special case). Finally one goes on to see if there is a solution to $dy/dx= y' = 0$. One finds in this case that there isn't but in general there might be.

 

[Mark44]

The question is :

Show that the curve defined implicitly by the equation

X*Y^3 + x^3*Y = 4

has no horizontal tangent.The answer is :

The above is not "the answer." It is the explanation of why the curve xy³ + yx³ = 4 doesn't have a horizontal tangent. The explanation seems pretty straightforward to me. What part of the explanation seems confusing to you?

Can someone please explain what steps are used in order to obtain the answer in a simple way for a calculus beginner? This answer is not helping me understand the question and how to answer the question ( It is too complicated and gets me confused).

And if you have similar questions, please show me them so I can practice on them after I understand this question in order to be comfortable with this type of questions.

 

[HallsofIvy]

Perhaps a little bit simpler: a "horizontal tangent" is where y'= dy/dx= 0. Differentiating both sides of $xy^3+ x^3y= 4$ with respect to x,
$y^3+ 3xy^2y'+ 3x^2y+ x^3y'= 0$.

At any point where y'= 0, that becomes $y^3+ 3x^2y= y(y^2+ 3x^2)= 0$. That is only true where y= 0 or $y^2+ 3x^2= 0$ which, again, requires that y= 0. But if y= 0 then $xy^3+ x^2y= 0$, not 4. That curves does not contain any point where y= 0 so no point where y'= 0.

 

[FactChecker]

I think that the original explanation is pretty good and thorough, step-by-step. It uses a lot of fundamental facts. You should look at it carefully step-by-step and ask about the specific steps that are a problem.