That's a pretty standard "Lagrange multipliers" problem. We want to minimize \(\displaystyle f(x,y,z)= 2x^2+ y^2+ 3z^2\) subject to the constraint \(\displaystyle g(x, y, z)= 2x- 3y-4z= 49\). A max or min will occur where \(\displaystyle \nabla f\) and \(\displaystyle \nabla g\) are parallel. That is, there exist a number, \(\displaystyle \lambda\) (the "Lagrange multiplier") such that \(\displaystyle \nabla f= \lambda\nabla g\).
Do you know what \(\displaystyle \nabla f= \nabla 2x^2+ y^2+ 3z^2\) is? Do you know what \(\displaystyle \nabla 2x- 3y- 4z\) is?
