91Ó°ÊÓ

Solve the following problems by the method of Lagrange multipiiers. a. Find the point on the plane \(a x+b y+c z=d\) that is nearest the origin. Hint: Minimize \(w=x^{2}+y^{2}+z^{2}\) with the side condition \(a x+b y+c z-d=0\) b. Show that the triangle with greatest area \(A\) for a given perimeter is equilateral. Hint: If \(x, y\), and \(z\) are the sides, then \(A=\) \(\sqrt{s(s-x)(s-y)(s-z)}\) where \(s=(x+y+z) / 2\). c. If the sum of \(n\) positive numbers \(x_{1}, x_{2}, \ldots, x_{n}\) has a fixed value \(s\), prove that their product \(x_{1} x_{2} \cdots x_{n}\), has \(s^{\prime \prime} / n^{n}\) as its maximum value, and conclude from this that the geometric mean of \(n\) positive numbers can never exceed their arithmetic mean: $$ \sqrt[n]{x_{1} x_{2} \cdots x_{n}} \leq \frac{x_{1}+x_{2}+\cdots+x_{n}}{n} $$