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【问题】(武汉市2007年高三二月调考理科数学第21题)巳知函数f(x)=x~2+2x+alnx.(1)若函数f(x)在区间(0,1]上恒为单调函数,求实数a的取值范围;(2)当t≥1时,不等式f(2_t—1)≥2f(t)—3恒成立,求实数a的取值范围.此题主要考查利用导数知识作工具,研究函数的单调性,处理不等式恒成立问题,综合性强,思想方法深刻,能力要求较高.其中第(2)小题难度较大,考生的答题情况并不理想.现就此小题的解法分析如下.
[Problem] (Wuhan City in 2007 in March and February tuition science Mathematics Question 21) Know the function f (x) = x ~ 2 + x + alnx. (1) If the function f (x) in the interval (0, 1] Constant constant is a monotonic function, and seeks the range of the real number a; (2) When t≥1, the inequality f(2_t-1)≥2f(t)-3 is established and the range of the real number a is obtained. This topic mainly examines the use of derivative knowledge as a tool to study the monotonicity of functions, deal with inequalities and constant establishment problems, comprehensiveness, deep thinking, and high ability requirements. Among them, the second (2) sub-question is more difficult, the candidate’s answer It is not ideal. The solution to this problem is analyzed as follows.