※ 引述《victor7935 (victor)》之铭言： 原 PO 真的有想过问题 ? 这些题目是作业吧? 问题咀嚼过後有问题再发问吧 : 1. Show that : x - (x^3)/6 < sinx for all x > 0 : 微1次後 : f(x) = sinx -x + (1/6)x^3 : f'(x)= cosx +1 + (1/2)x^2 : 然後..... Note that f is continuous and f(0) = 0 . f'(x) > 0 for every x . : 2. : Suppose that f is continuous on [a,b] and f(a)=f(b). : Show that f has at least one critical point in (a,b). f is continuous on [a,b] => f attains its maximum M & minimum m on [a,b] If f has no critical point in (a,b) , then M = m ( ∵ f(a) = f(b) ) , i.e. f is a const. function on [a,b] →← : 3. : (a) : Give an example of a nonconstant function that takes on both its : absolute max and absolute min on every interval. 取 Dirichlet function D(x) 即可 : (b) : Give an example of a nonconstant function that has an iinfinite : number of distinct local max and an infinite number of distinct : local min. x , x 属於 Z and x is even f(x) = { -x , x 属於 Z and x is odd 0 , x 属於 R\Z : 4.Use : 1 - (1/2)x^2 < cosx < 1- (1/2)x^2 + (1/24)x^4 for all x>0. : to estimate cos 6度(the x above is in radians.) π 6 ° = ------ rad ≒ 0.105 rad 30 => cos 6 ° ≒ 0.989 --

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