a^2/b + b^2/a >= a + b + 1
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Step-by-step explanation:
△=
∣
∣
∣
∣
∣
∣
∣
∣
1+a
2
−b
2
2ab
2b
2ab
1−a
2
+b
2
−2a
−2b
2a
1−a
2
−b
2
∣
∣
∣
∣
∣
∣
∣
∣
Operating R
1
→R
1
+bR
3
,
=
∣
∣
∣
∣
∣
∣
∣
∣
1+a
2
+b
2
2ab
2b
0
1−a
2
+b
2
−2a
−b(1+a
2
+b
2
)
2a
1−a
2
−b
2
∣
∣
∣
∣
∣
∣
∣
∣
Taking (1+a
2
+b
2
) common from R
1
,
=(1+a
2
+b
2
)
∣
∣
∣
∣
∣
∣
∣
∣
1
2ab
2b
0
1−a
2
+b
2
−2a
−b
2a
1−a
2
−b
2
∣
∣
∣
∣
∣
∣
∣
∣
Operating R
2
→R
2
−aR
3
,
=(1+a
2
+b
2
)
∣
∣
∣
∣
∣
∣
∣
∣
1
0
2b
0
1+a
2
+b
2
−2a
−b
a(1+a
2
+b
2
)
1−a
2
−b
2
∣
∣
∣
∣
∣
∣
∣
∣
Taking (1+a
2
+b
2
) common from R
2
,
=(1+a
2
+b
2
)
2
∣
∣
∣
∣
∣
∣
∣
∣
1
0
2b
0
1+a
2
+b
2
−2a
−b
a(1+a
2
+b
2
)
1−a
2
−b
2
∣
∣
∣
∣
∣
∣
∣
∣
Now, on expanding along R
1
, we get
=(1+a
2
+b
2
)
2
[(1−a
2
−b
2
)+(2a
2
+2b
2
)]
=(1+a
2
+b
2
)
2
.(1+a
2
+b
2
)
=(1+a
2
+b
2
)
3
.
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