二次不等式的問題

cellmembrane

給定f(x)=kx^2 +10x -5 及g(x)= 5x^2 +kx +k兩個函數,其中k >0

a)求k的可能值的範圍,使對於所有實數x, f(x)<= g(x).

請問條題目有咩hints比左我地?
我只係睇到 k>0 同埋f(x) <= g(x)
之後就列條式出黎,但係有k同埋x兩個未知數.請問應該點做?

THX!

Ivanhy92

f(x) <= g(x)

g(x)-f(x) >=0

(5-k)x^2 + (k-10)x + (k+5) >= 0  , which is a quadratic equation in x.

For a quadratic curve lies above or touches the x-axis,
the quadratic equation must have 1 real root or 0 real root.
Therefore, the leading coefficient must be positive and its discriminant must be non-positive.

Hence, from the given conditions, we require
k>0  and 5-k>0  and discriminant <=0
i.e. 0<k<5 and (k-10)^2-4(5-k)(k+5) <=0
etc.

之後自己solve吧
0 < k =< 4
成題數關鍵在於 g(x)-f(x)都係一條quadratic curve