<PRE><B>Question 538235</B>
Take a number, x. If it adds to some other number to equal 150, then 150-x is that other number.&lt;P&gt;
We have x and 150-x, and their product must be as large as possible.&lt;P&gt;
{{{x(150-x)=-x^2 + 150x}}}&lt;P&gt;
Graph it, or remember that it will be a parabola opening down.&lt;P&gt;
Thus, when its derivative equals 0 {{{-2x+150=0}}} the function has its maximum.&lt;P&gt;
Subtract 150 from both sides.&lt;P&gt;
{{{-2x = -150}}}&lt;P&gt;
Divide both sides by -2&lt;P&gt;
{{{x-75}}}&lt;P&gt;
The product of the numbers {{{x*(150-x)}}} has its maximum value when x=75.  And when x=75 then 150-x = 150-75 = 75.&lt;P&gt;
The two numbers are 75.&lt;P&gt;
You could also solve this by looking at a much smaller number, like 6.&lt;P&gt;
What two real numbers whose sum is 6 have the maximum product?  &lt;P&gt;
Compare 3 and 3 with 1 and 5 with 4 and 2 with 6 and 0.  The largest product is 3*3.&lt;P&gt;
Look at 5.  The largest product comes from 2.5 * 2.5&lt;P&gt;
For 8 the largest product comes from 4*4.  &lt;P&gt;
By observation you can see that when the sum is a positive number, then the largest product comes from the number divided by 2.&lt;P&gt;
150/2=75.&lt;P&gt;
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