Wednesday, January 13, 2010

Maximum value of a variable

#Problem 9
a+b+c+d+e=8
a^2+b^2+c^2+d^2+e^2=16. [a,b,c,d,e are real numbers]

Find the maximum value of e.

Solution Scheme and approach:
This problem can easily be done with Cauchy-Schwartz  inequality. But will explain this in my next topic. Let's do it without Cauchy.

(a-r)^2+(b-r)^2+(c-r)^2+(d-r)^2+(e-r)^2
=(a^2+b^2+c^2+d^2+e^2)-2r(a+b+c+d+e)+5r^2
=16-16r+5r^2[ By putting the values]

we can write
(e-r)^2<=16-16r+5r^2
=>(e-r)=root[16-16r+5r^2] [equality holds when a=b=c=d=r]
=>e=root[16-16r+5r^2]+r=f(r)
as this function is increasing we have only local minima
f'(r)=0
=>(10r-16)/root(16-16r+5r^2)=0...(i)
=>r=2 or 6/5
but r=2 is extraneous solution.Does not satisfy eqn number (i)
So r=6/5
=>e is maximum when a=b=c=d=6/5
=>max(e)=f(6/5)=16/5.


 

2 comments: