a+b+c=30. Find P=a*b*c such that?

Question

a) P is maximum
b) P is minumum

(a,b,c = intergers)

give general method for these type of questions also. -thanks.

Answer 1

P has max of 1000 when a=b=c=10
P has minimum of - infinity. Pick any arbitrarily large negative value for c. Then pick -c+10 for b and pick 20 for a. p will then be any arbitrarily large negative number. For example, let the value of c be -10^1000. then b = 10^1000 +10 and a =20.
There sum is 30, but P = -10^1000(10^1000 +10)20. So we can make the negative number approach infinity.

Answer 2

When all else fails, use method of exhaustion, listing all possible combinations of a, b, and c constrained by summation equaling 30 to produce product of a, b, and c, then check list for max and min. If you know how to write in BASIC or some other computer language, the solution to produce the list and find min/max is trivial since computer does all the work (a darn good reason to learn Excel or how to program your calculator).

Other methods exist in first-semester Calculus texts on finding min/max of functions by using and interpreting derivatives.

Answer 3

Pabc is maximum when all three numbers are the same (10,10,10) = 1000

Try it with any other triplet and you will see.
12*8*10 = 960
5*2*20 = 200

Pabc is a minimum when one (or two) of the numbers are 0. The product will be 0.

Answer 4

the other guy is correct
numbers to be maximum
1) product of two numbers - use the square root or geometric mean
2) in a sum of numbers - use the division or use arithmetic mean
3) in projectile motion - use the 45 degree inclination