1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Fimd the maximum value

  1. Nov 4, 2012 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Let a,b,c be real numbers such that a+2b+c=4. Then the max value of (ab+bc+ca) is

    2. Relevant equations

    3. The attempt at a solution
    I am trying to substitute for b into the second expression but that won't give me the answer. Any help would be appreciated.
  2. jcsd
  3. Nov 4, 2012 #2


    User Avatar
    2017 Award

    Staff: Mentor

    The approach is fine, what do you get as result?
  4. Nov 4, 2012 #3
    Note that a + c = 4-2b
    and ab + bc + ac is the same as b(a + c) + ac

    So you can write X = b(4-2b) + ac.
    That's a quadratic that you can differentiate wrt b to find a maximum for X at a particular b. You can also find what ac must be from the same result.

    After that, a few substitutions gives you the answer.
    Last edited: Nov 4, 2012
  5. Nov 4, 2012 #4


    User Avatar
    Gold Member

    Now substituting this into the second expression and simplifying I get

    Whats next?
  6. Nov 4, 2012 #5


    User Avatar
    Gold Member

    Here's what I did

    Differentiating wrt b and setting it to 0
    Now , ac=c(2-c)
    Diff again wrt c and setting it to 0
    Substituting a,b and c I get maximum value = 3

    But the answer is 4. I can't find out my mistake. Maybe there is something wrong with differentiation.
  7. Nov 4, 2012 #6


    User Avatar
    Science Advisor

    The derivative with respect to a is 2- a and the derivative wth respect to c is 2- c. Set those equal to 0 and you get a= c= 2 which then gives b= 0.
  8. Nov 6, 2012 #7


    User Avatar
    Gold Member

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook