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Find A \enspace and \enspace B so that the function f(x,y) = x^2+Ay +y^2 +B has a local...

Question:

Find {eq}A \enspace and \enspace B {/eq} so that the function {eq}f(x,y) = x^2+Ay +y^2 +B {/eq} has a local minimum at the point (0,4) with z-coordinate 35.

Graphing Functions; Local Extreme Points:

To find the values of {eq}A \text{ and } B {/eq} we'll set up two equations. One equation is given by the condition that {eq}f(0,4)=35. {/eq} The second equation comes from the fact that at {eq}(0,4) {/eq} the function {eq}f {/eq} has a local minimum, so the gradient of {eq}f {/eq} is the zero vector at that point. Solving the equations for {eq}A \text{ and } B {/eq} we get the desired values.

Answer and Explanation:

The fact that the {eq}z {/eq}-coordinate equals {eq}35 {/eq} at {eq}(0,4) {/eq} means that

{eq}35= 0^2+A(4) +(4)^2 +B, \text{ so we have that ...

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