# Finding Tha maxima and Minima of a Function given a constraint

If $$a_i = 1 - \frac{1}{N_i}$$ and $$\sum\limits_{I=0}^k{N_i} = n$$, then what is the maximum and minimum values of $$\sum\limits_{I=0}^k{a_i}$$?

Please help, I've tried to solve it but then I got confused. I think I may of found the minimum value to be $$\frac{n - k}{n - k + 1}$$ but I'm not sure.

Also $$N_i > 0$$ and $$N_i$$ is a subset of $$\mathbb{N}$$.

Note by Harry Obey
2 years, 2 months ago

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$$\displaystyle \sum_{i=0}^{k}a_{i} = k - \displaystyle \sum_{i=0}^{k}\dfrac{1}{N_{i}}$$
Using AM-GM-HM inequality,
$$\dfrac{\displaystyle \sum_{i=0}^{k}N_{i}}{k} \ge \dfrac{k}{\displaystyle \sum_{i=0}^{k}\dfrac{1}{N_{i}}}$$
$$\displaystyle \sum_{i=0}^{k} \dfrac{1}{N_{i}} \ge \dfrac{k^{2}}{n}$$
$$-\displaystyle \sum_{i=0}^{k} \dfrac{1}{N_{i}} \le - \dfrac{k^{2}}{n}$$
$$\displaystyle \sum_{i=0}^{k} a_{i} \le k - \dfrac{k^{2}}{n}$$
This is the maximum value of the expression, I am not sure about the minimum.

- 2 years, 2 months ago

Thanks so much, I have been trying to solve this problem for about 3 days. I heard that it may be possible to find the minimum using Lagrange multipliers but I'm not sure.

- 2 years, 2 months ago