Chapter 14.8, Problem 20E

To determine

We need to find the maximum of $f\left(x,y\right)=2x+5y$ on the ellipse ${\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)}^2+{\left(\raisebox{1ex}{$y$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)}^2=1$ without using Lagrange multipliers. First, show that the ellipse is parametrized by $x=4\cos t,y=3\sin t$. Then find the maximum value of $f\left(4\cos t,3\sin t\right)$ using single-variable calculus. Is one method easier than the other?