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Mathematics
Problem
By considering the function
y
=
(
a
1
x
−
b
1
)
2
+
(
a
2
x
−
b
2
)
2
y = (a_1x - b_1)^2 + (a_2x - b_2)^2
y
=
(
a
1
x
−
b
1
)
2
+
(
a
2
x
−
b
2
)
2
, use quadratic theory to prove the
Cauchy-Schwarz inequality
∣
a
1
b
1
+
a
2
b
2
∣
≤
a
1
2
+
a
2
2
b
1
2
+
b
2
2
|a_1b_1 + a_2b_2| \le \sqrt{a_1^2 + a_2^2} \sqrt{b_1^2 + b_2^2}
∣
a
1
b
1
+
a
2
b
2
∣
≤
a
1
2
+
a
2
2
b
1
2
+
b
2
2
.
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