## anonymous 5 years ago Find two numbers whose sum is 34 and whose product is a maximum.

1. anonymous

is there a graph that goes with this?

2. anonymous

No :S

3. anonymous

make a guess, i bet it will be right.

4. anonymous

isn't we have to do integral ? it's like calculus :D

5. anonymous

you need nothing but either simple (more or less) algebra, or common sense. do you know how to find the vertex or a parabola?

6. anonymous

You can think about this as a graph I guess. Let x,y be the numbers we are searching for, than x+y=34 We want to find the max of x*y=x(34-x) from the above equation. where does this have a max, you can find it by differentiating it as a function and getting 34-2x=0 x=17 but I guess x cannot be =y so the answer is 16-18.

7. anonymous

what do you think sat?

8. anonymous

Andras is right. but you can also just say the graph of $y=x(34-x)=34x-x^2$ has vertex at $-\frac{b}{2a}=-\frac{34}{-2}=17$

9. anonymous

or you can do none of the above and use common sense. call the numbers a and b, and you know a + b = 34. you are looking for the maximum of ab. but a + b = b + a, that is you cannot tell them apart. (symmetric in a and b) so it is obviously biggest when they are equal. just like the area of a rectangle with fixed perimeter is largest when you make a square.

10. anonymous

but if you are taking a calculus course, then you clearly should write Andras method!

11. anonymous

This is a grade 10 math course... and I was wondering what a maximum was when you don't have a parabola?