Question

if X + Y is equal to 120 the maximum value of the product of xy is​

Answer 1

Answer:

3600

Step-by-step explanation:

The value of X + Y is 120.That means that as we approach towards the maximum value where both the numbers are equal is 60.And hence 60*60 will give 3600

Answer 2

Given that the value of x + y is 120. We have to find the maximum value of xy.

Answer is simply 3600.

The shortcut is that let x = y = 60, since 60 is half of 120.

But what would be the logic behind this shortcut?!

If we're given two numbers, then we can write that two numbers as the sum and difference of any two particular numbers respectively or vice versa.

Applicable in this case too. We can write x and y in the form of u + v and u - v respectively or vice versa.

Let me consider this situation. Let x = u + v and y = u - v.

In terms of x and y, we can derive u and v too.

u = (x + y) / 2

v = (x - y) / 2

But, given that x + y = 120. So 'u' will be,

u = (x + y) / 2  =  120 / 2 = 60

Thus, in this case u = 60. So x = 60 + v and y = 60 - v.

Now, what about xy?!

xy = (60 + v)(60 - v) = 3600 - v²

The logic of the shortcut is due to this one!

Consider xy = 3600 - v².

From this we can have that, if the value of v in the equation increases, then so will be v², but the value of xy decreases faster, because of the RHS.

So, if xy decreases when v increases, then xy will be higher when v is smaller, i.e., vice versa.

But what about if v decreases more becoming a negative number?!

Even if it does, v² won't decreases to negative, because we know that the square of any number is always non-negative.

So, importantly, in this case we don't get mistaken by considering value of v, actually we have to consider '|v|', i.e., modulus of v, or absolute value of v.

Now, let me rewind what I said!

Consider xy = 3600 - v². If |v| is greater, then xy will be lesser. So xy will be higher if |v| is smaller.

So, xy will be the highest if |v| is the least, right?!

Now, we know modulus of a number is just its magnitude only. Only that it's non-negative.

So isn't the least value of modulus of a number 0?!

Yes, thus let |v| = 0. This is the least value of |v|. It's taken to get the maximum value of xy.

So,

xy = 3600 - 0 = 3600.

That's why the maximum value of xy is 3600.

We just used the shortcut based on this!

The fact is, if we're given sum of two numbers, then the square of half of that sum will be the maximum value of the product of the two numbers.

Algebraically, if  x + y = k,  then the maximum value of xy = (k / 2)².