Find the absolute maximum and minimum of the function fxy X

Find the absolute maximum and minimum of the function f(x,y) = X2 + y2 subject to the constraint x4 + y4 = 16. As usual, ignore unneeded answer blanks, and list points in lexicographic order. Absoluta minimum value: attained at Absolute maximum value: attained at

Solution

using the constraint: x^4 + y^4 = 16 ==> y^4 = 16 - x^4 ==> y = +/- (16 - x^4)^(1/4) plug in to f(x,y) to make it f(x) = x^2 + (16 - x^4)^(1/2) f \' = 2x - 4x^3 / 2(16 - x^4)^(1/2) = 2x - 2x^3 / (16 - x^4)^(1/2) f \' = 0 when 2x = 2x^3 / (16 - x^4)^1/2 x(16 - x^4)^(1/2) = x^3 (16 - x^4)^(1/2) = x^2 16 - x^4 = x^4 16 = 2x^4 8 = x^4 x = +/- 8^(1/4), or the four root of 8, or +/- 2^(3/4) when x = +/- 2^(3/4), then y = +/- 2^(3/4), and x^2 + y^2 = 2 * 2^(3/2) = 2(8^(1/2)) = 4sqrt(2) absolute max = 4sqrt(2) absolute min will occur at the boundaries of the domain and range: (+/- 2 , 0) and (0, +/- 2) here, f(xy) = 4