Wednesday, January 13, 2010

Maximum value of a variable

#Problem 9
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.

=16-16r+5r^2[ By putting the values]

we can write
=>(e-r)=root[16-16r+5r^2] [equality holds when a=b=c=d=r]
as this function is increasing we have only local minima
=>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