Mysterious Maxima

Algebra Level 2

$A$ and $B$ are two positive real numbers such that $A\times B=100$ .
What is the maximum value of $A+B$ ?

20 25 101 The maximum value does not exist

2 solutions

William Whitehouse
Sep 18, 2016

It is not specified that A and B are integral; as such we can choose an arbitrarily large number $n$ and have $A=100n$ and $B=\frac{1}{n}$ .

The sum can therefore take any value greater than or equal to 20.

Applying AM-GM inequality it is clear that upper bound of A+B doesn't exist.

Deepak Kumar - 4 years, 8 months ago

Nope. You have only shown that a minimum value exists, but you didn't show that an upper bound fails to exist.

Pi Han Goh - 4 years, 8 months ago

Came up with that 2, but then i Let me trick myself into thinking A and B must be integers lees then 100... don't have a club why i thought that haha

Peter van der Linden - 4 years, 8 months ago

If we take 100 and 1, the sum of A+B is 101....Doesn't it the greatest??????????????????

shithil Islam - 4 years, 6 months ago

Why can't A = 10000, B = 0.01? The sum is A+B = 10000.01 is greater than your answer

Pi Han Goh - 4 years, 6 months ago

Oh, it was not in my mind that a positive real number can be (.) Thank you so much to remembered me.....;)

shithil Islam - 4 years, 6 months ago

Tina Sobo
Oct 15, 2016

If AB=100, then B=100/A and the sum, A + B = A + 100/A = (101/100)*A. There is no bound on A in the positive direction, so as A becomes larger, the sum of A+B will get larger (and, incidentally, B will approach 0, and the sum will approach A), but A could be 5 trillion 42 and the sum would be 5 trillion 42 and a smidge. I was worried that the maximum value might approach something, because I don't remember limits well, but since the problem is multiple choice, it is clear that the sum can be larger than 101, therefore the maximum value must not be defined.

Mysterious Maxima

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