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    Hi, last question! See attached file.

    Using the sphere as a constraint I reduced to 2 variables:

    f(x,z) = xz + 4 - x^2 - z^2

    Using df/dx=0 and df/dz=0 I deduced x=0, z=0...giving points (0,2,0) and (0,-2,0).

    I calculated the value of DELTA = -3 < 0.
    Since f_xx = -2 < 0 I concluded that both points are maxima...

    Can anyone kindly verify this? Should I have used Langrange multipliers?
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    Your function is continuous on a closed and bounded domain, so it must have a minimum value. And it clearly isn't constant, so the minimum isn't also a maxima.

    So I think there must be at least 1 minimum value you haven't found.

    I think the problem is going to be that any such minimum is going to be attained on the boundary of the (x,z) circle, so your approach of eliminating a variable and differentiating isn't going to work.

    For a simpler case, suppose you want to maximize x^2-y^2 subject to x^2+y^2 = 1. It's obvious the solution is x=+/-1. But if you eliminate y, you get y^2 = 1-x^2, so x^2-y^2 = 2x^2 - 1, and then setting df/dx = 0 gives you x = 0; you completely miss the solutions at the constraint boundaries.
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    Is my method for obtaining maxima (0,2,0) and (0,-2,0) ok ?

    For the boundary points I should check (2,0,0) (-2,0,0) (0,0,2) and (0,0,-2) ? All give the value of f(x,y,z) as zero. So these points are minima?
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    (Original post by vc94)
    Is my method for obtaining maxima (0,2,0) and (0,-2,0) ok ?
    Think so, but I'm a bit rusty on all this.

    For the boundary points I should check (2,0,0) (-2,0,0) (0,0,2) and (0,0,-2) ? All give the value of f(x,y,z) as zero. So these points are minima?
    No, you need to check every point on the boundary. When x=\sqrt{2}, z=-\sqrt{2} you get a -ve value for f (which I suspect will be the minimum value but I haven't actually checked).

    N.B. I haven't done any actual calculus looking at this question, just looked at the actual form of the function to make some educated guesses about where it will be big or small.

    Edit: I think the above is rubbish - I didn't notice the '4' constant. So I think your answers are probably correct, but you should still check the whole boundary I think.
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    Bit confused...the point (root 2, 0, -root 2) on the sphere gives the value of as -2. But the point isn't a stationary point so can't be a minimum??
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    In some crude sense, it isn't. If you keep going outwards from the circle in the x-z plane, you'll get even lover values. For instance f(2, 0, -2) is even lower. The catch is only that in these cases y is imaginary, and the problem is probably posed with the silent assumption that imaginary numbers aren't allowed. (Otherwise, you could get as small numbers as you like by choosing x, z with large absolute values, one neagative and one positive). So, just differentiating isn't enough here, since you are constrained to the closed disk x^2 + z^2 <= 4 and you wish to find the global minimum within this region, global minimum aren't necessarily local minimums, and you need to check the whole boundary circle, as DFranklin says.
 
 
 

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