# Find two positive real numbers such that they add to 52 and their product is as large as possible.?

### 6 Answers

- SumDudeLv 72 months ago
TWO makes me think 25 and 27 (if they must be whole numbers). Otherwise 26 (minus a tiny fraction) and 26 (plus the same tiny fraction). [Calculus gives the proof that a number squared always results in the largest number (area, etc).

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- PhilipLv 62 months ago
Let numbers be x, 52-x.

Their product = f(x) = 52x - x^2.= = x(52-x).

f'(x) = 52 - 2x.

f''(x) = -2.

An extremum occurs at f'(x) = 0, ie., at x = 26.

Since f''(x) < 0, any extremum reached will be

a maximum.

f(26) = 26(52-26) = 26^2 = 4*13^2 = 4*(170-1)

= 676. Both pos. real numbers = 26 and f(x) reaches its maximum at (26,676).

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- sepiaLv 72 months ago
Find two positive real numbers such that they add to 52

and their product is as large as possible.

Let the two positive real numbers be x and y

x + y = 52

xy = 26^2

(52 - y)y = 26^2

y^2 - 52y + 26^2 = 0

y = 26, x = 26

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- llafferLv 72 months ago
Let x and y = your two positive real numbers.

The sum is 52, so:

x + y = 52

Let's solve for y in terms of x:

y = 52 - x

You want the largest product possible, so we can set up this function using two unknowns:

f(x, y) = xy

We have y in terms of x so we can substitute to make the function have one unknown:

f(x) = x(52 - x)

f(x) = 52x - x²

The "x" that gives the maximum can be found by solving for the zero of the first derivative:

f'(x) = 52 - 2x

0 = 52 - 2x

2x = 52

x = 26

Now that we have x, we can solve for y:

y = 52 - x

y = 52 - 26

y = 26

The largest product you can get from two real numbers with the sum of 52 is:

xy

26(26)

676

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- az_lenderLv 72 months ago
26 and 26. Note that

x^2 > (x-1)(x+1) > (x-2)(x+2) > (x-3)(x+3) > and so forth.

So you could put ANY even number into your question,

and the answer would always be, split it in half.

- DixonLv 72 months agoReport
Nice, there's always one more trick using the difference of two squares :-)

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- DixonLv 72 months ago
If there is a maximum (and by inspection, there is) then by symmetry it must be when the two numbers are equal., ie 26 and 26.

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