Let x, y, z be real numbers such that 4x^2 + y^2 + 16z^2 = 1. Find the maximum value of 7x + 2y + 8z.
+9519 By Lagrange multipliers,
\(\begin{cases} \nabla_{x,y,z}{\left(4x^2 + y^2 + 16z^2 - 1\right)} = \lambda \nabla_{x,y,z}\left(7x + 2y + 8z\right)\\ 4x^2 + y^2 + 16z^2 = 1 \end{cases}\)
\(\begin{cases} x = \dfrac78 \lambda\\ y = \lambda\\ z = \dfrac14\lambda\\ 4x^2 + y^2 + 16z^2 = 1 \end{cases}\)
Substitute the first three equations to the fourth one,
\(\dfrac{49}{16}\lambda^2 + \lambda^2 + \lambda^2 = 1\\ \lambda^2 = \dfrac{16}{81}\\ \lambda = \pm \dfrac 49\)
Optimum occurs when \((x,y,z) = \left(\dfrac7{18}, \dfrac49, \dfrac19\right) \) or \((x,y,z) = \left(-\dfrac7{18}, -\dfrac49, -\dfrac19\right) \)
Substitute these values to find that the maximum value of 7x + 2y + 8z occurs when \((x,y,z) = \left(\dfrac7{18}, \dfrac49, \dfrac19\right) \), and the value is \(\dfrac92\).
