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函数是中学数学的重要内容.没有给出具体解析式的函数,由于它将具体函数的性质高度抽象化,因此使不少同学望而生畏,束手无策.解这类题要求我们思维灵活,通过联想具体函数的有关性质,探索解题方法. 一、线性函数 例1 已知函数f(x)的定义域是R,对任意x1,x2∈R都有f(x1十x2)=f(x1)+f(x2),且当x>0时,f(x)<0,f(1)=a,试判断在区间[-3,3]上,f(x)是否有最大值或最小值,如果有,求出最大值或最小值,如果没有,说明理由. 分析虽然求函数最值方法很多,但本题是函数的抽象,只能利用函数的单调性求解,由条件易联想教材中的函数f(x)=kx,进而证明f(x)在R上是递减求解.
The function is an important part of middle school mathematics. No concrete analytical function is given. Because it abstracts the nature of specific functions, many students are daunted and helpless. Solving such questions requires us to be flexible in thinking through the association of concrete functions. The nature of the problem, explore the solution to the problem. First, linear function Example 1 The domain of the known function f (x) is R, for any x1, x2 ∈ R have f (x1 ten x2) = f (x1) +f (x2), and when x>0, f(x)<0, f(1)=a, try to determine whether f(x) has a maximum or minimum value in the interval [-3,3] if Yes, find the maximum or minimum value, if not, explain the reason. Although the analysis of the method to find the most value of the function is a lot, but this problem is an abstraction of the function, can only use the monotony of the function to solve, from the conditions easily think of the function of the textbook f (x) = kx, which proves that f(x) is a degressive solution to R.