Problems with two constraints Given a differentiable function \(w=f(x, y, z),\)
the goal is to find its absolute maximum and minimum values (assuming they
exist) subject to the constraints \(g(x, y, z)=0\) and \(h(x, y, z)=0,\) where \(g\)
and \(h\) are also differentiable.
a. Imagine a level surface of the function \(f\) and the constraint surfaces
\(g(x, y, z)=0\) and \(h(x, y, z)=0 .\) Note that \(g\) and
\(h\) intersect (in general) in a curve \(C\) on which maximum and minimum values
of \(f\) must be found. Explain why \(\nabla g\) and \(\nabla h\) are orthogonal to
their respective surfaces.
b. Explain why \(\nabla f\) lies in the plane formed by \(\nabla g\) and \(\nabla
h\) at a point of \(C\) where \(f\) has a maximum or minimum value.
c. Explain why part (b) implies that \(\nabla f=\lambda \nabla g+\mu \nabla h\)
at a point of \(C\) where \(f\) has a maximum or minimum value, where
\(\lambda\) and \(\mu\) (the Lagrange multipliers) are real numbers.
d. Conclude from part (c) that the equations that must be solved for maximum
or minimum values of \(f\) subject to two constraints are \(\nabla f=\lambda
\nabla g+\mu \nabla h, g(x, y, z)=0,\) and \(h(x, y, z)=0\)
