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To find the maximum and minimum values of a function f(x, y) subject to the constraint g(x, y) = 0, the method of Lagrange multipliers can be used. Geometrically, it can be shown that the maximum (minimum) value of f will occur where the level curves of f and the level curves of g share a common tangent line. At this point the gradient of f 1 and the gradient of g will be parallel and f ( x , y) = g( x , y) . It follows that f x ( x , y) = g x ( x , y) f y ( x , y) = g y ( x , y) Using these equations, together with g(x, y) = 0, can be eliminated and the values of x and y corresponding to the maximum and minimum values of f can be determined. The next example illustrates the procedure.
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EXAMPLE 7 Suppose we wish to find the maximum and minimum values of f(x, y) = 2x2 + 3y2 subject to the
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Eliminate[equations, k] eliminates between a set of simultaneous equations. See 6.
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constraint x2 + y2 = 4. We define g(x, y) = x2 + y2 4 and eliminate . f[x_, y_]= 2 x + 3 y ; g[x_, y_]= x2 + y 2 4;
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conditions = Eliminate[{ xf[x, y] k x g[x, y], yf[x, y] k y g[x, y], g[x, y] 0}, k ] x 2 4 y 2 & & x y 0 & & 4y + y 3 0 points = Solve[conditions] {{x 2, y 0}, {x 0, y 2}, {x 0, y 2}, {x 2, y 0}} To determine the maximum and minimum values of f, we compute its values at these points. functionvalues = f[x, y] /. points {8, 12, 12, 8} Max[functionvalues] 12 Min[functionvalues] 8 The method of Lagrange multipliers can be extended to functions of three (or more) variables.
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EXAMPLE 8 To find the maximum and minimum values of f(x, y, z) = xyz, subject to the constraint x2 + 2 y2 + 3 z2 = 6,
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we define g(x, y, z) = x2 + 2 y2 + 3 z2 6. f[x_, y_,z_]= x y z;
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g[x_, y_,z_]= x2 + 2 y 2 + 3 z2 6; conditions = Eliminate[{ xf[x, y,z] k x g[x, y,z] , yf[x, y,z] k y g[x, y,z], zf[x, y,z] k z g[x, y,z], g[x, y,z] 0}, k] x2 6 2 y 2 3 z2 & & 4 y 2 z z(6 3 z2)& & x(2 y 2 3 z2) 0 & & x y( 2+ 3 z2) 0 & & x z( 2+ 3 z2) 0 & & y z( 2+ 3 z2) 0 & & 4 z 8 z3 + 3 z5 0 & & y 3 + y( 3 + 3 z2) 0 points = Solve[conditions]
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{{x 0, y 0, z 2 },{x 0, y 0, z 2 }, {x 0, y 0, z
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2 },{x 0, y 0, z 2 },{x 0, y 3 , z 0},
The gradient of f ( x, y) is the vector function f ( x, y) = f x ( x, y) i + f y (x, y) j. The gradient of f ( x, y, z) is f ( x, y, z) = f x ( x, y, z) i + f y (x, y, z) j + f z ( x, y, z) k.
Multivariate Calculus
{x 0, y 3 , z 0},{x 0, y
3 , z 0},{x 0, y 2 3 },
3 , z 0},
2 {x 2 , y 1, z 3 },{x 2 , y 1, z 2 {x 2 , y 1, z 3 },{x 2 , y 1, z
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2 , y 1, z 2 },{x 2 , y -1, z 3 2 2 , y 1, z 2 },{x 6 , y 0, z 0}, 2 , y 1, z 3 3 },{x
2 3 }, 2 3 },
{x 6 , y 0, z 0},{x 6 , y 0, z 0},{x 6 , y 0, z 0}}
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