Find two numbers whose sum is 34 and whose product is a maximum.

Question

anonymous · Answer

is there a graph that goes with this?

anonymous · Answer

No :S

anonymous · Answer

make a guess, i bet it will be right.

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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.

anonymous · Answer

what do you think sat?

anonymous · Answer

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

anonymous · Answer

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.

anonymous · Answer

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

anonymous · Answer

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