Chapter 13.9, Problem 33ES

Let $\alpha ,\beta ,\text{and}\gamma$ be the angles of a triangle.

(a) Use Lagrange multipliers to find the maximum value of $f\left(\alpha ,\beta ,\gamma \right)=\cos \alpha \cos \beta \cos \gamma ,$ and determine the angles for which the maximum occurs.

(b) Express $f\left(\alpha ,\beta ,\gamma \right)$ as a function of $\alpha ,\text{and}\beta$ alone, and use a CAS to graph this function of two variables. Confirm that the result obtained in part (a) is consistent with the graph.