Find the maximum value of the function f(x, y, z) = x + 4y + 9z on the curve of intersection of the plane x-y + z = 1 and the cylinder x2 + y2 = 1. SOLUTION we maximize the function x, y, z) = x + 4y + 9z subject to the constraints g(x, y, z)-x-y + z = 1 and h(x, y, z) = x2 + y2-1. The Lagrange condition is Vf= λ▽g + ㎶h, so we solve the equations 4="-A + 2yu (3) 9 x-y+z= 1 Putting λ = 9 Similarly, (2) gives y = 13/(21). Substitution in (5) then gives [from (3)] in (1), we get 241 =-8 16 and so tr=233/4 , μ = ±v/233/2. Then x = 干! 233 21 V233The 13 and, from (4), z = 1-x + y = 1 ± corresponding values of fare 13 Therefore the maximum value of f on the given curve is 9 V 233

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