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\(\displaystyle \displaystyle \begin{align*} \nabla f(x, y) &= \lambda \nabla g(x, y) \\ g(x, y) &= k \end{align*}\)

yet? Here \(\displaystyle \displaystyle f(x, y) = x^2y\) and \(\displaystyle \displaystyle g(x, y) = x^2 + 2y^2 = 6\).

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Yeah I tried. I ended up with 2y^2+sqrt(2)y-6 which doesn't seem right and if it is right, I don't know how to solve it

\(\displaystyle \displaystyle \begin{align*} \nabla f(x, y) &= \lambda \nabla g(x, y) \\ g(x, y) &= k \end{align*}\)

yet? Here \(\displaystyle \displaystyle f(x, y) = x^2y\) and \(\displaystyle \displaystyle g(x, y) = x^2 + 2y^2 = 6\).

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Hello again, GWR309!f(x,y)=x^2y with the constraint of x^2+2y^2=6

Use lagrange multipliers to find the extrema.

Thanks!

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Sorry but that's not even close. Start by evaluating the gradient vector for each of those functions. Remember that \(\displaystyle \displaystyle \nabla f(x, y) = \left( \frac{\partial f }{\partial x} , \frac{\partial f}{\partial y} \right)\)Yeah I tried. I ended up with 2y^2+sqrt(2)y-6 which doesn't seem right and if it is right, I don't know how to solve it

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The objective function is:

\(\displaystyle f(x,y)=x^2y\)

subject to the constraint:

\(\displaystyle g(x,y)=x^2+2y^2-6=0\)

Now, first find the implications of the system:

\(\displaystyle f_x(x,y)=\lambda g_x(x,y)\)

\(\displaystyle f_y(x,y)=\lambda g_y(x,y)\)

Then use the implications in the constraint to find the critical points. Can you write down the system from which to take the implications?

Which if we wrote as vectors would look likeProve Itis being more rigorous, is to write:

The objective function is:

\(\displaystyle f(x,y)=x^2y\)

subject to the constraint:

\(\displaystyle g(x,y)=x^2+2y^2-6=0\)

Now, first find the implications of the system:

\(\displaystyle f_x(x,y)=\lambda g_x(x,y)\)

\(\displaystyle f_y(x,y)=\lambda g_y(x,y)\)

Then use the implications in the constraint to find the critical points. Can you write down the system from which to take the implications?

\(\displaystyle \displaystyle \begin{align*} \left[ \begin{matrix} f_x (x, y) \\ f_y (x, y) \end{matrix} \right] &= \lambda \left[ \begin{matrix} g_x(x, y) \\ g_y (x, y) \end{matrix} \right] \\ \nabla f(x, y) &= \lambda \nabla g(x, y) \end{align*}\)