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I have 2 input $X$ and $Y$ which are both positive integers. I have to maximize this function

Let $A=\min(Y/4,X/2)$ , $B=\min(W/2,Y/2)$, $C=\max(A,B)$, and $D=\max(X-W,Y)$.

Then $$ f(X,Y)=\max_{0\leq W\leq X} C\cdot C\cdot D $$

How can this problem be solved? Is it possible to determine whether this function will have a single global maximum or not?

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  • $\begingroup$ W can vary from 0 to X . $\endgroup$ Apr 9 '13 at 21:14
  • $\begingroup$ @julien:W varies from 0 to X . $\endgroup$ Apr 9 '13 at 21:15
  • $\begingroup$ @julien:Real values. $\endgroup$ Apr 9 '13 at 21:26
  • $\begingroup$ So is my edit correct? $\endgroup$
    – Julien
    Apr 9 '13 at 21:28
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The easy part is this: If $Y\ge 2X$, then $A=X/2$, $B=W/2$, $C=B=W/2$, $D=X-W$, hence $$f(X,Y)=\max_{0\le W\le X} \frac{W^2(X-W)}4=\left.\frac{W^2(X-W)}4\right|_{W=\frac23X}=\frac{X^3}{27}.$$ But if $X\le Y<2X$, you should try to find $$ f(X,Y)=\max\left\{\max_{0\le W\le Y/2}C^2D, \max_{Y/2< W\le X}C^2D\right\}$$ and the case $Y<X$ need still a bit more consideration.

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  • $\begingroup$ is the function f(X,Y)monotonically increasing /decreasing for X<=Y<2X $\endgroup$ Apr 10 '13 at 13:22

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