Minimizing $2a^2 + b^2 + c^2$ given $4a + 3b + c = 7$

Question

Problem: Find the minimum value of the expression $2a^2 + b^2 + c^2$ if the point $(a,b,c)$ lies on the plane $4a + 3b + c = 7$.

I couldn't get very far, but I do know how would I solve a simpler problem, i.e. minimizing $a^2 + b^2 + c^2$ as that would simply be the perpendicular distance from the origin to the plane. I tried re-writing the expression as

$$ (a^2 + b^2) + (a^2 + c^2) $$ In hopes of finding something. But here also nothing significant was achieved. Any hints are appreciated!

Answer 1

Hint,

Let $f(a,b,c)=2a^2+b^2+c^2$ and $g(a,b,c)=4a+3b+c-7$ ,

Now consider $\nabla f=\lambda \nabla g$ with $g(a,b,c)=0$.

Answer 2

By application of Cauchy-Schwarz Inequality

$$ \left( \sum_{i=1}^{n} a_i ^2 \right)\left( \sum_{i=1}^{n} b_i ^2 \right) \geq \left( \sum_{i=1} a_i b_i \right)^2$$

We have $$ (2a^2 + b^2 + c^2)((4/\sqrt{2})^2 + 3^2 + 1^2) \geq ((4/\sqrt{2} (\sqrt{2}a)+3b+c)^2 $$ $$ (2a^2 + b^2 + c^2)(18) \geq (4a+3b+c)^2 $$ $$ 2a^2 + b^2 + c^2 \geq 49/18 $$

Hence the minimum value of the given expression is $49/18$.