1.
(x^2+y^2)/2 ≧ xy
xy ≦ 2/2 = 1
2.
a > 0
(a+1/a)/2 ≧ √(a)(1/a) = 1
a + 1/a ≧ 2
a < 0
a + 1/a = -((-a) +(-1/a))
(-a) +(-1/a) ≧ 2
-((-a) +(-1/a)) ≦ -2
a + 1/a ≦ -2
3.
μ= (1+2+3+4+5)/5 = 3
f(x)之最小值 = f(3) = 4+1+0+1+4 = 10
4.
x-1=(y+1/2)=(z-2/3) ??
假設為 x – 1 = (y+1)/2 = (z-2)/3
x = t + 1, y = 2t – 1, z = 3t + 2
x^2+y^2+z^2
=(t-1)^2 + (2t-1)^2 + (3t+2)^2
= 15t^2 + 6t + 6
=15(t^2 + 2t/5) + 6
=15(t+1/5)^2 + 6 – 15(1/25)
=15(t+1/5)^2 + 27/5
x^2+y^2+z^2之最小值為 27/5
5.
x^4+2x^2+5 = (x^2)^2 + 2(x^2) + 5 = (x^2 + 1)^2 +5 – 1
x^2 = 0, 最小值 = 1 + 5 -1 = 5
6.
x^2+2y^2=1, y^2 = (1-x^2)/2
1 – x^2 ≧ 0
-1 ≦ x ≦ 1
2x+3y^2
= 2x + 3(1-x^2)/2
= -3x^2/2 + 2x + 3/2
=(-3/2)(x^2 – 4/3x) + 3/2
=(-3/2)(x – 2/3)^2 + 3/2 – (-3/2)(4/9)
=(-3/2)(x – 2/3)^2 + 13/6, -1 ≦ x ≦ 1
x = 2/3, 2x+3y^2之最大值為 13/6
x= -1, 2x+3y^2之最小值為 (-2/3)(-1-2/3)^2 + 13/6 = -2
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