find a pair of integer whose sum is _100
Answers
Answer:
Suppose two numbers sum to equal
100
. Let
x
represent the first number. Then the second number must be
100
−
x
, and their product must be
x
(
100
−
x
)
=
−
x
2
+
100
x
.
As
f
(
x
)
=
−
x
2
+
100
x
is a downward opening parabola, it has a maximum at its vertex. To find its vertex, we put it in vertex form, that is,
a
(
x
−
h
)
2
+
k
where
(
x
,
f
(
x
)
)
=
(
h
,
k
)
is its vertex.
To put it into vertex form, we use a process called completing the square:
−
x
2
+
100
x
=
−
(
x
2
−
100
x
)
=
−
(
x
2
−
100
x
)
−
(
100
2
)
2
+
(
100
2
)
2
=
−
(
x
2
−
100
x
)
−
2500
+
2500
=
−
(
x
2
−
100
x
+
2500
)
+
2500
=
−
(
x
−
50
)
2
+
2500
Thus the vertex is at
(
x
,
f
(
x
)
)
=
(
50
,
2500
)
, meaning it attains a maximum of
2500
when
x
=
50
.
As such, the pair of numbers
x
,
100
−
x
attains a maximal product when
x
=
50
, meaning the desired pair is
50
,
100
−
50
, or
50
,
50
.