Find the minimum product of two numbers who difference is 12. What are the two numbers? (How did you get it?)

Question

anonymous · Answer

3 and 4 (I looked at the multiplication table)

anonymous · Answer

is that it?

anonymous · Answer

is it -11

anonymous · Answer

How Matricked?

anonymous · Answer

because the answer is -36 and the two numbers are -6, 6

anonymous · Answer

but you said product, right? So I don't get it....Sorry

anonymous · Answer

product is multiply

anonymous · Answer

I know and that's why I'm asking :(

anonymous · Answer

oops my ans is incorrect yes it may be -36

anonymous · Answer

Now I am totally confused...........Sorry

anonymous · Answer

but how? and why?

anonymous · Answer

are you sure?

anonymous · Answer

yes. The question asks for: FInd the minimum product of two numbers whose difference is 12 and then answer is -36. Also, what are the two numbers: -6, 6

anonymous · Answer

6, -6

anonymous · Answer

How, A.M $\ge $ GM

anonymous · Answer

A.M? GM?

anonymous · Answer

arithmetic mean and geometric mean respectively.

anonymous · Answer

put one number x, the other is x + 12
minimize 
$$x(x+12)=x^2+12x$$ vertex is 
$$(-6,-36)$$

anonymous · Answer

ohh so how does solve -6, 6 if the vertex is -6,-36?

anonymous · Answer

satellite satellite !!! :)

anonymous · Answer

find vertex via 
$$x=-\frac{b}{2a}$$

anonymous · Answer

two numbers are x and x+ 12
we find x = -6 and therefore we know other one is -6 + 12 = 6

anonymous · Answer

ohh... how did you come up with x+12?

anonymous · Answer

ok lets imagine one number is 5 and the difference is 12
what is the other number?

anonymous · Answer

huh? so it is 7? o.o

anonymous · Answer

i get 17 because 17 - 5 = 12
how did i get it? added 12 to 5

anonymous · Answer

Btw  A.M vs G.M generally works for non-negative numbers but (some what surprisingly) works here

anonymous · Answer

ohh.. i get it alright foodformath :)

anonymous · Answer

put one number x, the other is x + 12 minimize
Let Product p = x(x+12)=x2+12x

use Maxima Minima method to find the least product.
differentiate the function
p' = 2x + 12
for maxima or minima p' =0 
=> 2x + 12 =0 
=> x=-6 ............. the first no.
=> x+12 = +6 .... the second no.

 as f'' = 2 ........ a positive no. so the function has a minima at x= -6 and not a maxima.