MHB Find Min Value of Expression: $\prod_{i=1}^{2017}x_i$

  • Thread starter Thread starter lfdahl
  • Start date Start date
  • Tags Tags
    Value
lfdahl
Gold Member
MHB
Messages
747
Reaction score
0
If $x_1,x_2,...,x_{2017} \in\Bbb{R}_+$
and $\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_{2017}} = 1$
- then find the minimal possible value of the expression: \[\prod_{i=1}^{2017}x_i\]
 
Mathematics news on Phys.org
My solution:

By cyclic symmetry, we know the critical value is at the point:

$$\left(x_1,\cdots,x_{2017}\right)=(2016,\cdots,2016)$$

And the objection function at that point is:

$$f(2016,\cdots,2016)=2016^{2017}$$

Now, looking at another point on the constraint:

$$\left(4032,4032,\cdots,4032,\frac{2016}{2017}\right)$$

We find the objective function at that point is:

$$f\left(4032,4032,\cdots,4032,\frac{2016}{2017}\right)=\frac{2^{2016}2016^{2017}}{2017}>2016^{2017}$$

And so we conclude:

$$f_{\min}=2016^{2017}$$
 
MarkFL said:
My solution:

By cyclic symmetry, we know the critical value is at the point:

$$\left(x_1,\cdots,x_{2017}\right)=(2016,\cdots,2016)$$

And the objection function at that point is:

$$f(2016,\cdots,2016)=2016^{2017}$$

Now, looking at another point on the constraint:

$$\left(4032,4032,\cdots,4032,\frac{2016}{2017}\right)$$

We find the objective function at that point is:

$$f\left(4032,4032,\cdots,4032,\frac{2016}{2017}\right)=\frac{2^{2016}2016^{2017}}{2017}>2016^{2017}$$

And so we conclude:

$$f_{\min}=2016^{2017}$$

Thankyou, MarkFL for your correct solution!:cool:
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
2
Views
1K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
22
Views
5K
Replies
4
Views
1K
Back
Top