[?] Maximize a function created by adding two exponential functions?

Questions
,
1

Hello, I had a 3D function which consisted of a sum of two quadratic funtions:

Maximize[x1*(600 - 4*x1) + x2*(340 - 1/3*x2),{x1,x2}]

which nicely yields the maximum of a concave paraboloid.
But when I need to maximize a function created by adding two exponential functions, I do not get a solution although the answer is simple.

Maximize[x1*Exp[Log[600] - (4/600)*x1] + x2*Exp[Log[340] - (1/1020)*x2],{x1,x2}]

This is easily solvable on paper, I’m just wondering how to get WM to calculate this.

Thanks

2

One way is using NMaximize:

NMaximize[x1*Exp[Log[600] - (4/600)*x1] + x2*Exp[Log[340] - (1/1020)*x2], {x1, x2}]
(* {160690., {x1 -> 150., x2 -> 1020.}} *)

or:

Maximize[x1*Exp[Log[600.] - (4./600.)*x1] + x2*Exp[Log[340.] - (1/1020.)*x2], {x1, x2}]
(* {160690., {x1 -> 150., x2 -> 1020.}} *)
FindMaximum[x1*Exp[Log[600] - (4/600)*x1] + x2*Exp[Log[340] - (1/1020)*x2], {x1, x2}]
(* {160690., {x1 -> 150., x2 -> 1020.}} *)
3

It will also work if you give some contraints:

Maximize[{x1*Exp[Log[600] - (4/600)*x1] + 
   x2*Exp[Log[340] - (1/1020)*x2], {x1, x2} \[Element] 
   Rectangle[{0, 0}, {2000, 2000}]}, {x1, x2}]

gives

{436800/E, {x1 -> 150, x2 -> 1020}}
4

Thank you both very much! Very helpful, much appreciated! :slight_smile: