$$ \displaystyle \ \text{Prove} \ 3( a^2 + b^2 + c^2 + d^2) + 4abcd \geq 16 \ \text{subject to constraint} \ a+b+c+d=4 .$$
While I was going through this question Solving Inequality using Lagrange Multiplier, I became aware of this technique. So I tried to use it in this Question.
My Approach:
First I defined a constriant function $\displaystyle g(x) = a+b+c+d $ and a constraint set $$\displaystyle \bar{S} = \{ x \in \bar{U} : g(x) = 4 \}\\ \text{where} \ \bar{U} = \{ (a,b,c,d) : a+b+c+d \leq 100 \} $$
Now to the next step: Defining objective function $\displaystyle f(x) = 3( a^2 + b^2 + c^2 + d^2) + 4abcd .$
Also $g$ and $f$ are polynomial function, thus they are continuous.
Case $1:$
The minimum of $f$ lies on the boundary of the set $\bar{S}$, hence atleast one of the component of $\displaystyle g(x)$ is equal to zero.
Thus we are left to prove $\displaystyle 3( a^2 + b^2 + c^2 ) \geq 16 $ given that $a+b+c=4.$ This is the immediate result of Chauchy Swartz Inequality.
Case $2:$
Here we assume that minimum of $f$ lies inside set $\bar{S}$ $ ( \text{or} \ x \in S)$. Then using the Lagrange multiplier theorem:
The Lagrange Function here is $\displaystyle \mathcal{L}(x,\lambda) = f(x) - \lambda g(x)$. Thus to find local extrema of a Lagrange Function :
$$\displaystyle \nabla \mathcal{L}(x,\lambda) = 0 \implies \nabla f(x) = \lambda \nabla g(x) \\ $$
$$\displaystyle \left[ \begin{array}[c] *6a+ 4bcd \\ 6b + 4acd \\ 6c + 4abd \\ 6d + 4abc \\ \end{array} \right ] = \lambda \left[ \begin{array}[c] *1\\ 1\\ 1\\ 1\\ \end{array} \right ] $$
And the only case that satify this condition is the one with $a=b=c=d.$
Thus satisfying the constraint function $g(x) = 4,$ we get $a=b=c=d=1$ and then $ \displaystyle f(x) = 16 $.
And as the second partial derivative $\displaystyle \nabla^2 \mathcal{L}(x,\lambda) = 6,$ we conclude that $(1,1,1,1)$ is minimum of $f(x).$
Please correct me, if I am wrong or if there is something missing to check cause, I am new to this topic. And I want to ask if I am right: Lagrange function is the family of functions passing through the intersection of the two given functions (according to my knowlege of co-ordinate geometry).