Find the extreme values of f (x, y, z) = x^2 y z + 1 on the intersection of the plane z = 3 with...


Find the extreme values of {eq}f (x,\ y,\ z) = x^2 y z + 1 {/eq} on the intersection of the plane {eq}z = 3 {/eq} with the sphere {eq}x^2 + y^2 + z^2 = 12 {/eq}.


Typically finding the extrema of a multivariable function is anything but easy. However, given some restrictions, we can simplify things immensely. Since we have a function of three variables in the situation described above, and we have two restrictions, we will be able to reduce the function to only one variable. Then we can use our usual methods to find the extrema.

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Our restriction is that we are on the intersection of the plane and the sphere. This means everywhere in our equations, we can write {eq}z = 3 {/eq}....

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