<s>[INST] If $ 3a+2b+c=7$ then find minimum value of $ a^2+b^2+c^2$ Question:- If $ 3a+2b+c=7$ then find the minimum value of $ a^2+b^2+c^2 $.
I used vectors to solve this problem.
Let $$α=3\hat{i}+2\hat{j}+\hat{k}$$
$$β=a\hat{i}+b\hat{j}+c\hat{k}$$
Using Cauchy-Schwarz inequality
we have, $|α.β|\le |α| |β|$
$=|3a+2b+c|\le\sqrt{14}\sqrt{a^2+b^2+c^2}$
$= 7\le\sqrt{14}\sqrt{a^2+b^2+c^2}$
So, $a^2+b^2+c^2\ge \frac72$
Therefore, the minimum value of $a^2+b^2+c^2$ is $\frac72$
I want to know are there any other method to find the minimum value of
$a^2+b^2+c^2$ such as using inequalities and calculus by assuming function $f(x,y,z)=x^2+y^2+z^2$.
 [/INST] There is a variant of the Cauchy-Schwarz inequality and its name is by a certain community of problem solvers as Titu’s lemma:
$a^2+b^2+c^2 = \dfrac{(3a)^2}{9}+\dfrac{(2b)^2}{4}+\dfrac{c^2}{1} \ge \dfrac{(3a+2b+c)^2}{9+4+1} = \dfrac{49}{14} = \dfrac{7}{2}$.
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