# Find A and B so that f(x,y) = x^2 + Ax + y^2 + B has a local minimum at the point (6,0), with...

## Question:

Find {eq}A {/eq} and {eq}B {/eq} so that {eq}f(x,y) = x^2 + Ax + y^2 + B {/eq} has a local minimum at the point {eq}(6,0) {/eq}, with {eq}z {/eq}-coordinate 45.

## Partial Derivatives

To find the values of {eq}A {/eq} and {eq}B {/eq}, we first find the partial derivatives and their zeros. To find the partial with respect to {eq}x {/eq}, we treat {eq}y {/eq} as a constant and differentiate with respect to {eq}x. {/eq} Likewise to find the partial with respect to {eq}y {/eq}, we treat {eq}x {/eq} as a constant and differentiate with respect to {eq}y. {/eq}

{eq}f_x(x,y) = 2x + A \\ f_y(x,y) = 2y {/eq}

There is a minimum at the point {eq}(6,0) {/eq} therefore, this point is a critical point and both...

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