Question about max of functions

[alomari2010]

Let a,b,c,d in R. For a=< x =< b and c=< y =< d, find

max_x{max_y{(x-a)(y-c), (x-a)(d-y), (b-x)(y-c), (b-x)(d-y)}}??

it is well known that

max(x,y) = \frac{x+y+|x-y|}{2}

 

[Petr Mugver]

Let a,b,c,d in R. For a=< x =< b and c=< y =< d, find

max_x{max_y{(x-a)(y-c), (x-a)(d-y), (b-x)(y-c), (b-x)(d-y)}}??

it is well known that

max(x,y) = \frac{x+y+|x-y|}{2}

With x fixed the functions you have are just straight lines... very easy to find max and min!

max_x{max_y{(x-a)(y-c), (x-a)(d-y), (b-x)(y-c), (b-x)(d-y)}}=
max_x{(x-a)(d-c), (x-a)(d-c), (b-x)(d-c), (b-x)(d-c)}=
max{(b-a)(d-c), (b-a)(d-c), (b-a)(d-c), (b-a)(d-c)}==(b-a)(d-c)

 

[alomari2010]

Thanks!

but i think you didn't got what i want!

i will try to repost my Q?

As above we have a formula for max{x,y}. So what is the formula for max{x,y,z}?
where x,y,z in R. In general, for x_i in R, i=1,2,..,n
what is the formula for max{x_1, x_2,..., x_n}?

hope anyone can got the answer!