Page 97 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I inequAlities

√
√ x+ y
3.30 If x, y > 0, find the maximum value of √ x+y .
√
√
√ x+ y x+y+2 xy 2 xy 2 xy √
√
√
2
Solution: f(x, y)= √ x+y , f (x, y)= x+y = 1+ x+y ≤ 1+ 2 xy =2 , thus f(x, y) ≤ 2,
√
√
√ x+ y √
which means the maximum value of √ x+y is 2.
5
3.31 Given x> 0, show x + 1 + 1 1 ≥ .
x x+ 2
x
1
1
1
1
1
Solution: Let f(x) = x + (x> 0), then x + 1 ≥ 2. Let 2 ≤ α<β, then f(α) − f(β)= (α + ) − (β + ) =(α − β)+ ( − ) = (α−β)(αβ−1) < 0
x x α β α β αβ
1
1
1
1
f(α) − f(β)= (α + ) − (β + ) =(α − β)+ ( − ) = (α−β)(αβ−1) < 0 , that is, f(x) is an increasing function
α β α β αβ
on [2, +∞). Hence, f(x + ) ≥ f(2) = .
5
1
x 2
3.32 Real numbers a, b, c satisfy a + b + c =0, abc =2, show that at least one of a, b, c
is not less than 2.
Proof: Obviously at least one of a, b, c is positive. Without loss of generality, let a> 0,
then b + c = −a, bc =2/a , that is, b, c are the two roots of the quadratic equation
2
2
3
x + ax + 2 =0. Consider the discriminant Δ ≥ 0 ⇒ a − 8 ≥ 0 ⇒ a ≥ 8 ⇒ a ≥ 2.
a a

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