Section 14.7: Max Min Problems

Theorem If \(f\) has a local maximum or minimum at \((a, b)\) and the first-order partial derivatives of \(f\) exist there, then \(f_x(a, b) = 0\) and \(f_y(a, b) = 0\).

PROOF Let \(g(x) = f(x, b)\). If \(f\) has a local maximum (or minimum) at \((a, b)\), then \(g\) has a local maximum (or minimum) at \(a\), so \(g'(a) = 0\) by Fermat’s Theorem. But \(g'(a) = f_x(a, b)\) and so \(f_x(a, b) = 0\). Similarly, by applying Fermat’s Theorem to the function \(G(y) = f(a, y)\), we obtain \(f_y(a, b) = 0\).

Notice that the conclusion of Theorem 2 can be stated in the notation of gradient vectors as \(\nabla f(a, b) = \mathbf{0}\).

If we put \(f_x(a, b) = 0\) and \(f_y(a, b) = 0\) in the equation of a tangent plane, we get \(z = z_0\). Thus the geometric interpretation of Theorem 2 is that if the graph of \(f\) has a tangent plane at a local maximum or minimum, then the tangent plane must be horizontal.