the complete question is
Find two numbers whose difference is 46 and whose product is a minimum
Let
x------->larger number
y-------> smaller number
P-------> product of the two numbers
we know that
[tex] x-y=46\\ y=x-46 [/tex]-----> equation 1
[tex] P=x*y [/tex]-----> equation 2
substitute equation 1 in equation 2
[tex] P=x*[x-46]\\ P=x^{2} -46x [/tex]
using a graph tool
see the attached figure
Find the value of x for that the product P is a minimum
the vertex is the point [tex] (23,-529) [/tex]
that means, for [tex] x=23 [/tex]
the product is a minimum [tex] P=-529 [/tex]
find the value of y
[tex] y=x-46\\ y=23-46\\ y=-23 [/tex]
therefore
the answer is
the numbers are [tex] 23 [/tex] and [tex] -23 [/tex]
