Determine the minimum value of #a/(2b) + b/(4c) + c/(8a)# where #a,b,c# are positive real numbers?

2 Answers
May 9, 2017

#3/4#

Explanation:

Making #a = lambda_1 b# and #b=lambda_2 c# we have

#a/(2 b) + b/(4 c) + c/(8 a) =lambda_1/2 + 1/(8 lambda_1 lambda_2) + lambda_2/4 #

Now with #f(lambda_1,lambda_2) = lambda_1/2 + 1/(8 lambda_1 lambda_2) + lambda_2/4 #

and determining the stationary points with

#{((partial f)/(partial lambda_1)=1/2 - 1/(8 lambda_1^2 lambda_2)=0),((partial f)/(partial lambda_2)=1/4 - 1/(8 lambda_1 lambda_2^2)=0):}#

and solving for #lambda_1, lambda_2# we get the real values

#lambda_1 = 1/2, lambda_2 = 1#

Those values are relative to a local minimum for #f(lambda_1,lambda_2)# because at that point the hessian

#grad^2 f =1/2 ((1,1),(1,4))# with characteristic polynomial

#p(s)=3/4 - (5 s)/2 + s^2=(s-1/4 (5 - sqrt[13]))(s-1/4 (5 + sqrt[13]))#

have two positive roots, qualifying the point as a minimum point.

The value attained for #f# at this point is #3/4#

May 9, 2017

# (a/(2b)+b/(4c)+c/(8a))_(min)=3/4.#

Explanation:

We use the AMGM Inequality (AG) to solve this Problem.

The AG Property :

# AA x,y,z in RR^+, (x+y+z)/3 ge root(3)(xyz).#

Applying this to #x=a/(2b), y=b/(4c), z=c/(8a)," all "gt0," we have,"#

#1/3{a/(2b)+b/(4c)+c/(8a)} ge root(3){a/(2b)*b/(4c)*c/(8a)}.#

#rArr 1/3{a/(2b)+b/(4c)+c/(8a)} ge root(3)(1/64)=1/4.#

#rArr a/(2b)+b/(4c)+c/(8a) ge 3/4.#

#:. (a/(2b)+b/(4c)+c/(8a))_(min)=3/4,# as derved by Respected

Cesareo R., Sir!

Enjoy Maths.!