f.4 inequality

用戶9285

2009-07-05 6:22 am

1. let α and β e the roots of the equaion (x-1)^2=k^2-k+2, where k is a real number
a)show that α and β are real and distinct
b) find the minimum value of ︱α-β︳

2. α and β are the roots of the equation x^2-(m-1)x+3(m-5)= 0 where m is real.
a) show that α and β are real and distinct
b) show that (3-α)(3-β)

3. let α and β be the roots of the quadratic equation x^2-(m+3/m)x+2=0where m is a non-zero constaant
a)using the fact that (a-b)^2≧0 for all real numbers a and b , show thatm^2+9/(m^2)≧6
hence, show that (m+3/m)^2≧12

b) using the result of (a), deduce that α and β are distinct real numbers
c)find the range of values of m such that ︱α+β︳≦2αβ

更新1:

3. a) where is (m-3/m)^2 come from?

回答 (1)

用戶9112

2009-07-05 7:41 am

✔ 最佳答案

1.a)(x-1)^2=k^2-k+2
= x^2 -2x -(k^2 - k +1)

D = 4 + 4(k^2 - k +1)
= 4k^2 - 4k + 8
=(2k-1)^2 + 7>0
Therefore the roots are real and distinct

b) The two roots are [2+/-sqrt(D)]/2
︱α-β︳= sqrt(D) which is a minimum when k=1/2
and sqrt(D) = sqrt(7)

2.a) D= (m-1)^2 - 4*3(m-5)
=m^2 - 2m + 1 - 12m + 60
=m^2 -14m + 61
=(m+7)^2 + 12 > 0
Therefore the roots are real and distinct

b) (3-α)(3-β)
= 9+αβ- 3(α+β)
= 9 + 3(m-5) - 3(m-1)
= 9 + 3m - 15 - 3m + 3
= -3

3. a)(m-3/m)^2 = m^2 - 6 + 9/m^2 >= 0
so m^2 + 9/m^2 >=6

(m+3/m)^2 = m^2 + 6 +9/m^2 >= 6 + 6 =12

b) D = (m+3/m)^2 - 8 >= 12-8 = 4 >0
Therefore the roots are real and distinct

c)α+β= (m+3/m)
2αβ= 4
︱α+β︳<= 2αβ means (α+β)^2 <= (2αβ)^2
(m+3/m)^2<=16
m^2 + 6 + 9/m^2 <=16
m^2 -10 + 9/m^2 <=0
(m-9/m)(m-1/m)<=0

Case (i) m>0
(1) m^2-9>=0 and m^2-1<=0; or
(2) m^2-9<=0 and m^2-1>=0

(1) m>=3 and m<=1; or
(2) m<=3 and m>=1

So 1<=m<=3

Case (ii) m<0
(1) m^2-9>=0 and m^2-1<=0; or
(2) m^2-9<=0 and m^2-1>=0

(1) m<=-3 and m>=-1; or
(2) m>=-3 and m<=-1

So -3<=m<=-1

2009-07-05 21:19:50 補充：
Discriminant = b^2 - 4ac

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