please help with part C Consider the curve x²y + y²x = 6A) Find in terms of x and ydxB) Write the equation for the tangent line where x = 2 and y = 1.C) Find the coordinates of the point (x, y) where the curve has a horizontal tangent line.

Answered: Consider the curve x²y + y?x = 6 A)…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 43E

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please help with part C

Consider the curve x²y + y²x = 6
A) Find in terms of x and y
dx
B) Write the equation for the tangent line where x = 2 and y = 1.
C) Find the coordinates of the point (x, y) where the curve has a horizontal tangent line.

Expert Solution

Step 1

c). The given function is:

$x^{2} y + y^{2} x = 6$

For finding the coordinates of point where the curve has a horizontal line:

So differentiate it:

$x^{2} y + y^{2} x = 6 \frac{d}{d x} (x^{2} y + y^{2} x) = \frac{d (6)}{d x} (2 x y + x^{2} \frac{d y}{d x}) + (2 x y \frac{d y}{d x} + y^{2}) = 0 \frac{d y}{d x} (x^{2} + 2 x y) = - (y^{2} + 2 x y) \frac{d y}{d x} = - \frac{(y^{2} + 2 x y)}{(x^{2} + 2 x y)}$

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