Question

The question

Answer 1

\(\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}}\)

Answer 2

20 i think

Answer 3

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

Answer 4

if im doin it right lol

Answer 5

So p,q,r are real numbers?

Answer 6

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...

Answer 7

that should be an approximation symbol there

Answer 8

\(\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}}\)

Answer 9

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

Answer 10

the correct option given by book is option \(c.)\)

the book also gave a hint to assume \(p=4,q=3,r=3\)

Answer 11

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

Answer 12

oh sry i forget they are integers given.

Answer 13

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

Answer 14

but book's solution is kind of trial and error

Answer 15

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

Answer 16

or is thaat the only way in case they are integers

Answer 17

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

Answer 18

http://www.wolframalpha.com/input/?i=maximize+x%5E4%2By%5E4%2Bz%5E4%2C+x%5E2%2By%5E2%2Bz%5E2%3D1%2Cx%3E0%2Cy%3E0%2Cz%3E0

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

using AM-GM even in case of integers ?

Answer 23

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

Answer 24

ok thnx.

Answer 25

69