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在判断函数的单调性和求函数的极值时,常常需要判断其导函数在某区间的符号,通常的方法是解不等式,但往往很麻烦困难。如例1 求函数f(x)=e~x+e~(-x)+2cosx的极值。解 f′(x)=e~x-e~(-x)-2sinx,解方程 e~x-e~(-x)-2sinx=0得唯一的驻点为x=0,此时f′(x)在x=0附近的函数值符号不易确定,需求高阶导数才能能判定f(x)在x=0处是否取极值。又如
When judging the monotonicity of the function and finding the extremum of the function, it is often necessary to judge the sign of its derivative function in a certain range. The usual method is to solve the inequality, but it is often troublesome and troublesome. As in Example 1, find the extremum of the function f(x)=e~x+e~(-x)+2cosx. The solution f′(x)=e~xe~(-x)-2sinx, the solution equation e~xe~(-x)-2sinx=0 has a unique stagnation point of x=0, and f’(x) is The sign of the function value near x=0 is not easy to determine, and the demand for higher order derivatives can determine whether f(x) takes the extreme value at x=0. Another example