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Relative maxima, relative minima, and saddle points

Find the relative maxima, relative minima, and saddle points of the function (x^2)*y - 6*(y^2) - 3*(x^2).

Solution Preview

Let f (x) = (x^2)*y - 6*(y^2) - 3*(x^2).

To determine relative minima/maxima and saddle points of f, we first find the gradient of f and set it to 0:

Let f_x and f_y denote the (first) partial derivatives of f with respect to x and y, respectively.

Then the x component of grad f is

f_x = (2x)*y - 0 - 3*(2x) = 2xy - 6x

and the y component of grad f is

f_y = (x^2) - 6*(2y) - 0 = (x^2) - 12y

If grad f = 0, then both f_x and f_y are 0.

To find points (x, y) at which grad f = 0, we solve the following system of equations:

Setting f_x to 0: 2xy - 6x = 0

Setting f_y to 0: (x^2) - 12y = 0

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To solve the first equation (2xy - 6x = 0), we factor the left-hand side:

2x(y - 3) = 0.

This implies that x = 0 or y = 3.

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Substituting each of these possibilities into the second equation (x^2 - 12y = 0) separately, we find the following:

If x = 0, then (x^2) - 12y = 0 implies that (0^2) - 12y = 0, which in turn implies that y = 0. Thus one possibility ...

Solution Summary

A complete, detailed solution is given, with an explanation of every step: It is determined whether the given function has any relative maxima (and if so, what they are), any relative minima (and if so, what they are), and any saddle points (and if so, what they are).

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