Implicit Differentiation Problem

[Vigo]

Consider the curve given by X^2+4y^2=7+3xy
a) show that dy/dx=3y-2x/8y-3x
b) show that there is a point P with x-cooridnate 3 at which the line tangent to the curve at P is horizontal. Find the y-cooridnate of P.
c)find the value of d^2y/dx^2 at the point P found in part (b). Does the curve have a local maximum, a local minimum, or neither at the point P? Justify your answer.

(a) is easy. All you do is find the derivative.

For (b), I got the point (3,2) by plugging 3 into the original equation and got 2.

For (c), the value I got was -2/7 by finding the second derivative and plugging (3,2) for the x's and y's. I need to know if this is right and if there are any max's or min's at this point. Thanks.

 

[0rthodontist]

-2/7 is correct. You know that dy/dx is 0 at the point, so you know it has to be either a maximum, a minimum, or a saddle point. Intuitively what do you think it should be, given that d^2y/dx^2 is less than 0? If it is less than 0 it means that the slope is decreasing--the slope is changing to become more negative. What would that mean in terms of maximum or minimum?

 

[HallsofIvy]

Slope of dy/dx is decreasing and is 0 at x= 3. That is, dy/dx is positive for x< 3 and negative for x> 3. y itself is increasing for x< 3, and decreasing for x>3. What does that tell you?

 

[Vigo]

Since y is increasing when x<3 and decreasing when x>3, does that mean there is a local maximum?

 

[HallsofIvy]

Yes. Draw a picture of that situation.

 

[Vigo]

Alright, thanks a lot for all of your help. I believe this graph looks like a slanted ellipse.