inequality

👨🏻‍🏫 學術台數學

Sean

2011-05-01 9:26 pm

Consider two quadratic functions f(x) = kx^2 + 10x - 5 and g(x) = 5x^2 + kx + k , where k > 0.

(a) Find the range of possible values of k such that f(x) ≦ g(x) for all real values of x.

(b) If the value of k found in (a) is an integer , find the possible values of k.

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更新1:

書既答案係 (a) 0 < k ≦ 4 (b) 1 , 2 , 3 , 4

更新2:

書既答案係 (a) 0 < k ≦ 4 (b) 1 , 2 , 3 , 4

更新3:

書既答案係 (a) 0 < k <= 4 (b) 1 , 2 , 3 , 4

回答 (1)

myisland8132

2011-05-01 9:49 pm

✔ 最佳答案

(a) f(x) ≦ g(x)

kx^2 + 10x - 5 ≦5x^2 + kx + k , where k > 0.

(5 - k)x^2 + (k - 10)x + (k + 5) > = 0

So, b^2 - 4ac <= 0 and 5 - k > = 0

For (k - 10)^2 - 4(5 - k)(k + 5) < = 0

k^2 - 20k + 100 + 4k^2 - 100 <= 0

k^2 <=4

-2 <= k <= 2

(b) k = -2,-1,0,1,2

2011-05-01 13:50:51 補充：
Sorry it should be k^2 - 4k <= 0

0 <= k <= 4

(b) 0,1,2,3,4

2011-05-01 18:33:34 補充：
當然書的答案是對的。這是因為 k > 0. 這些是minor mistakes 請見諒。

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