Find minimum of : \sum_{k=1}^{n}a_{k}^{2}+\left(\sum_{k=1}^n a_k\right)^2. My teacher told me that I shoul use Cauchy-Schwarz inequality. Any tips?

Question

Find minimum of : \sum_{k=1}^{n}a_{k}^{2}+\left(\sum_{k=1}^n a_k\right)^2.  My teacher told me that I shoul use Cauchy-Schwarz inequality. Any tips?

anonymous · Answer

$$ \sum_{k=1}^{n}a_{k}^{2}+\left(\sum_{k=1}^n a_k\right)^2$$
$$\left(\sum_{k=1}^n a\right)^2=\sum_{k=1}^n a^3$$
So your question boils down to
$$\sum_{k=1}^n a^2+ a^3=\sum_{k=1}^n a^2(a+1)$$

anonymous · Answer

http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32

anonymous · Answer

The link is the ()^2=(^3) proof

anonymous · Answer

Although that's not Cauchy Schwartz.

anonymous · Answer

I didn't realise that task has another assumption :$$\sum_{k=1}^{n}p_ka_k=1$$

anonymous · Answer

Sorry, I forgot that a1 does not necessarily equal 1, a2=2 etc. forget what I've told you so far.

anonymous · Answer

http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Statement_of_the_inequality you want the final formula of this section

anonymous · Answer

Or do you think a1=1, a2=2 etc in this problem?