## mathmath333 one year ago The question

1. mathmath333

\large \color{black}{\begin{align}& \{p,q,r\}\geq 0\hspace{.33em}\\~\\ & p+q+r=10 \hspace{.33em}\\~\\ & \normalsize \text{find the maximum value of} \ (pq+qr+pr+pqr)\hspace{.33em}\\~\\ \end{align}}

2. anonymous

20 i think

3. anonymous

because it 4 p's, 3 r's and 2 q's and you can only make two sets of (p=q=r)=10

4. anonymous

if im doin it right lol

5. freckles

So p,q,r are real numbers?

6. freckles

We know 70 something can be achieved since $\text{ if } p=q=r=\frac{10}{3} \\ \text{ then } 3 (\frac{10}{3})^2+(\frac{10}{3})^3=70.37$ but there might be a higher number we can reach then that maybe thinking...

7. freckles

that should be an approximation symbol there

8. mathmath333

\large \color{black}{\begin{align} & a.)\ \geq 40\ \cap \leq 50 \hspace{.33em}\\~\\ & b.)\ \geq 50\ \cap \leq 60 \hspace{.33em}\\~\\ & c.)\ \geq 60\ \cap \leq 70 \hspace{.33em}\\~\\ & a.)\ \geq 70\ \cap \leq 80 \hspace{.33em}\\~\\ \end{align}}

9. freckles

well if there is a bigger number than 70.37 and you have no other inequalities then process of elimination would mean...

10. mathmath333

the correct option given by book is option $$c.)$$ the book also gave a hint to assume $$p=4,q=3,r=3$$

11. freckles

that is assuming p,q,r are integers but that was never given

12. mathmath333

oh sry i forget they are integers given.

13. freckles

oh I would use the book's hint then :p

14. mathmath333

but book's solution is kind of trial and error

15. freckles

I think those numbers came from being close to 10/3

16. mathmath333

or is thaat the only way in case they are integers

17. ganeshie8

yeah but this symmetry stuff doesn't work always, remember we saw it failing multiple times before

18. ganeshie8
19. mathmath333

in case other question comes with restriction given as integers, should i use the same method as described by the book

20. ganeshie8

that method has no mathematical justification, simply saying "by symmetry" wont do

21. ganeshie8

you either need to use AM-GM inequality or other standard methods

22. mathmath333

using AM-GM even in case of integers ?

23. ganeshie8

it works on reals but you can use it to pick the closest integers

24. mathmath333

ok thnx.

25. dan815

69