数学题 数学高手进(高一函数)
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数学题 数学高手进(高一函数)
已知函数f(x)=x+ k/x
①若f(1)=2,试判断f(x)在(1,+∞)上的单调性,并给出证明.
②讨论f(x)在(1,+∞)上的单调性
(拜托各位,救命的啊!)
要用高一前两章的概念和名词哦!
呵呵...
(比如导数我就没听过)
已知函数f(x)=x+ k/x
①若f(1)=2,试判断f(x)在(1,+∞)上的单调性,并给出证明.
②讨论f(x)在(1,+∞)上的单调性
(拜托各位,救命的啊!)
要用高一前两章的概念和名词哦!
呵呵...
(比如导数我就没听过)
![数学题 数学高手进(高一函数)](/uploads/image/z/3436484-68-4.jpg?t=%E6%95%B0%E5%AD%A6%E9%A2%98+%E6%95%B0%E5%AD%A6%E9%AB%98%E6%89%8B%E8%BF%9B%EF%BC%88%E9%AB%98%E4%B8%80%E5%87%BD%E6%95%B0%EF%BC%89)
①
f(1) = x + k/x = 1 + k = 2
k = 1
f(x) = x + 1/x
设x1>x2>1
f(x1) - f(x2)
= (x1 - x2) + (1/x1 - 1/x2)
= (x1 - x2)·(x1·x2 - 1)/(x1·x2)
x1 > x2
x1·x2 > 1
所以
f(x1) - f(x2) > 0
f(x1) > f(x2)
在(1,+∞)上的单调递增
②
f(x) = x + k/x
设x1>x2>1
f(x1) - f(x2)
= (x1 - x2) + (k/x1 - k/x2)
= (x1 - x2)·(x1·x2 - k)/(x1·x2)
x1 - x2 > 0
x1·x2 > 1 >0
(1)
当k ≤ 1
x1·x2 - k > 0, f(x1) > f(x2), f(x)在(1,+∞)上的单调递增
(2)
当k > 1
当x1, x2 > √k
x1·x2 - k > 0, f(x1) > f(x2), f(x)在(√k,+∞)上的单调递增
当x1, x2 < √k
x1·x2 - k < 0, f(x1) < f(x2), f(x)在(1, √k)上的单调递减
f(1) = x + k/x = 1 + k = 2
k = 1
f(x) = x + 1/x
设x1>x2>1
f(x1) - f(x2)
= (x1 - x2) + (1/x1 - 1/x2)
= (x1 - x2)·(x1·x2 - 1)/(x1·x2)
x1 > x2
x1·x2 > 1
所以
f(x1) - f(x2) > 0
f(x1) > f(x2)
在(1,+∞)上的单调递增
②
f(x) = x + k/x
设x1>x2>1
f(x1) - f(x2)
= (x1 - x2) + (k/x1 - k/x2)
= (x1 - x2)·(x1·x2 - k)/(x1·x2)
x1 - x2 > 0
x1·x2 > 1 >0
(1)
当k ≤ 1
x1·x2 - k > 0, f(x1) > f(x2), f(x)在(1,+∞)上的单调递增
(2)
当k > 1
当x1, x2 > √k
x1·x2 - k > 0, f(x1) > f(x2), f(x)在(√k,+∞)上的单调递增
当x1, x2 < √k
x1·x2 - k < 0, f(x1) < f(x2), f(x)在(1, √k)上的单调递减