Question

确定常数a,b的值,使函数f(x)= 3sinx x？

Answer 1

因为分段函数f(x) = 3sinx,x < 0； f(x) = aln(1+ x) + b,x ≥ 0在x = 0处可导,所以函数f(x)在x = 0处连续：对x -> 0 - ,limf(x) =lim3sinx = 3*0 = 0 ； 对x -> 0 + ,limf(x) =limf(0) = aln1 + b = b ； 所以b = 0,f(x) = aln(1+ x),当x≥ 0 ； 1）当x < 0时,f(x) = 3sinx,求导可得f ’(x) = 3cosx,所以：对x -> 0 - ,limf ’(x) =lim3cosx = 3*1 = 3 ； 2）当x≥ 0时,f(x) = aln(1+ x),求导可得f ’(x) = a/(1+ x),所以：对x -> 0 + ,limf ’(x) =lim[a/(1 + x)] = a ； 所以a = 3 ； 综上所述,常数 a = 3 ,b = 0 .,5,