Cool Inequality #3 - Improved

Algebra Level 4

If $a_{i} > 0$ and $a_{1}^{2015}+a_{2}^{2015}+\cdots \cdots+a_{2015}^{2015}=2016$ . find the maximum value of $a_{1}+a_{2}+a_{3}..........+a_{2015}$ upto three decimal places.

The answer is 2014.999.

1 solution

Otto Bretscher
Sep 25, 2015

I'm using Lagrange multipliers to see that all the $a_k$ must be identical. Thus $2015a_k^{2015}=2016$ and the sum of all $a_k$ is just a little above $\boxed{2015}$ .

Can you tell me in detail?

Sarthak Singla - 5 years, 8 months ago

The gradients of the constraint function and the objective function must be parallel (the gradient being the list all the 2015 partial derivatives).

Now $grad(a_1,a_2,...,a_{2015})=(1,1,...,1)$ and $grad(a_1^{2015},...,a_{2015}^{2015})$ $=2015(a_1^{2014},...,a_{2015}^{2014})$ . It follows that $a_i^{2014}=a_j^{2014}$ for all $i$ and $j$ . Since the $a_k$ are positive, they must all be equal.

I will be glad to provide further details as needed.

Otto Bretscher - 5 years, 8 months ago

Cool Inequality #3 - Improved

The answer is 2014.999.

1 solution

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