Find the points on the ellipse 4x^2 + y^2 = 1 where f(x, y)...

Find the points on the ellipse $4x^2 + y^2 = 1$ where $f(x, y) = xy$ has its extreme values.\ Identify the following set of equations.\ $\nabla f = y\mathbf{i} + x\mathbf{j} = \lambda \nabla g$\ $\nabla g = 8x\mathbf{i} + 2y\mathbf{j}$\ $g(x, y) = 4x^2 + y^2 - 1 = 0$\ The points are

Transcript

00:01 In this problem, we want to find the points on the ellipse x square plus 4y square equal to 1, where fx and y is equal to xy, has its extreme values.

00:17 So essentially, what we want to do to find the extreme values is to find the critical points of our 2d function of x and y.

00:30 But since we have a constraint, which is the ellipse, we're going to reduce our 2d function, into a 1d function.

00:39 So first let's rewrite our ellipse equation as a condition on x and y.

00:57 So we have that x is equal to 1 minus 4 y square to the square root.

01:09 And let's insert this condition into our function f of x and y.

01:16 Now we have that our two -dimensional function can be written as a one -dimensional function solely as a function of y.

01:30 So we have the f of y is equal to y times d square root of 1 minus 4 y square.

01:48 So now to find the extreme values, we want to differentiate f with respect to y.

02:02 And then find a location for which this derivative is equal to 0.

02:08 So differentiating this expression, we obtain the square root of 1 minus 4 times y squared plus y times 1 half times 1 over 1 minus 4 y square to the square root times the derivative of 1 minus 4 y square which is equal to minus 8y so we see that our derivative simplifies to 1 minus 4 y square minus 4 times y square over d square root of 1 minus 4 y square.

03:15 So now we have a derivative to find the critical points...

Question

Find the points on the ellipse $4x^2 + y^2 = 1$ where $f(x, y) = xy$ has its extreme values.\ Identify the following set of equations.\ $\nabla f = y\mathbf{i} + x\mathbf{j} = \lambda \nabla g$\ $\nabla g = 8x\mathbf{i} + 2y\mathbf{j}$\ $g(x, y) = 4x^2 + y^2 - 1 = 0$\ The points are

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