Postby **prong** » Thu Sep 24, 2009 1:54 am

For 53, the expression doesn't depend on y at all; you maximize the possible configurations of x,z available to you when y = 0, so your condition can just be x^2 + z^2 <=2.

You're trying to find the minimum of f(x,y,z) = x + 4z. Extrema occur either where the derivative is 0 or on the boundary. In this case the derivative is grad(f) = (df/dx, df/dy, df/dz) [really those df/dx_i's are partial derivatives but I don't want to bother to pull out some LaTeX].

f is nice so grad(f) is nice, grad(f) = (1,0,4). this is never 0 so any extrema occur at the boundary of our region, i.e. x^2 + z^2 = 2. therefore x = -sqrt(2-z^2).

now you want to minimize x+4z = -sqrt(2-z^2) + 4z. this is the same as d/dz(-sqrt(2-z^2) + 4z) = 0, which is the same as -z/sqrt(2-z^2) + 4 = 0.

solving this, z/sqrt(2-z^2) = 4 so z^2/(2-z^2) = 16 and z^2 = 32-16z^2 gives us 17z^2 = 32. thus z = sqrt(32/17)

crap, did i screw up somewhere?