Lagrange Multiplier
Suppose a function \(f\) has an extreme value at a point \(P(x_0,y_0,z_0)\) on the surface \(S\).
Let \(C\) be a curve defined by a vector equation \(r(t) = \left\langle x(t), y(t), z(t) \right\rangle\) that lies on \(S\) and passes through \(P\).
It \(t_0\) is the parameter corresponding to \(P\), then
\[r(t_0) = \left\langle x_0, y_0, z_0 \right\rangle\]Then, the composition function
\[h(t) = f(x(t), y(t), z(t))\]represents the values that \(f\) takes on the curve \(C\). Since \(f\) has an extreme value at \((x_0,y_0,z_0)\), \(h\) has an extreme value at \(t_0\). Hence,
\[h'(t_0) = 0\]But if \(f\) is differentiable, we can apply the chain rule for \(h\),
\[h'(t_0)=0\] \[f_x(x_0,y_0,z_0)x'(t_0) + f_y(x_0,y_0,z_0)y'(t_0) + f_z(x_0,y_0,z_0)z'(t_0) = 0\] \[\nabla f(x_0,y_0,z_0) = r'(t_0)\]This shows that the gradient vector \(\nabla f(x_0,y_0,z_0)\) is orthogonal to the tangent vector \(r'(t_0)\) to every such curve \(C\).
But we know that \(\nabla g(x_0,y_0,z_0)\) is also orthogonal to the tangent vector \(r'(t_0)\) for every such curve \(C\).
Therefore, the two gradient vectors \(\nabla f(x_0,y_0,z_0)\) and \(\nabla g(x_0,y_0,z_0)\) must be parallel. So \(\nabla g(x_0,y_0,z_0) \neq 0\), there is a number \(\lambda\) such that
\[\nabla f(x_0,y_0,z_0) = \lambda \nabla g(x_0,y_0,z_0)\]The number \(\lambda\) is called a Langrage multiplier.
Method of Lagrange Multipliers
To find the maximum and minimum values of \(f(x,y,z)\) subject to the constraint \(g(x,y,z) = k\) assuming that these extreme values exist and \(\nabla g \neq 0\) on the surface \(g(x,y,z) = k\),
[i] Find all values of \(x,y,z,\lambda\) such that
\[\nabla f(x,y,z) = \lambda g(x,y,z)\] \[g(x,y,z) = k\][ii] Evaluate \(f\) at all the points \((x,y,z)\) that result from [i]. The largest value is the maximum value of \(f\) and the smallest value is the minimum value of \(f\).
Two Constraints
Tow constraints case work similarly
\[\nabla f(x_0,y_0,z_0) = \lambda \nabla g(x_0,y_0,z_0) + \mu \nabla h(x_0,y_0,z_0)\]References
[1] Stewart Calculus, 8th edition