4. 試應用均值定理(The Mean Value theorem),證明       cos⁡(b)-cos(a)≤ b-a, a,bR . or sin b -sin(a)≤ b-a, a,bR?

(cos(b)-cos(a))/(b-a) = -sin(u) 
         for som u between a and b

故, 對此 u
    cos(b) - cos(a) = -sin(u) (b-a)
若 b > a, 則 
    -sin(u)(b-a) ≦ b-a
若 a > b, 則
    -sin(u)(b-a) = sin(u)(a-b) ≧ -(a-b) = b-a
故 
    b > a 得 cos(b) - cos(a) ≦ b - a;
    a > b 則 cos(b) - cos(a) ≧ b - a,

(sin(b)-sin(a))/(b-a) = cos(v)
        for som v between a and b

故, 對此 v,
    sin(b) - sin(a) = cos(v)(b-a)
若 b > a 則 sin(b) - sin(a) ≦ b - a;
若 a > b 則 sin(b) - sin(a) ≧ b - a.


因此,
    cos(b) - cos(a) ≦ b - a 及
    sin(b) - sin(a) ≦ b - a
均成立於 all a, b such that a < b,
而非 for all a, b in R.