Question

已知函数f(x)=sinkx(sinx)^k+coskx(cosx)^k-(cos2x)^k

Answer 1

存在,f(x)连续可导

故要使f(x)为常数,则f'(x)=0恒成立

而f'(x)=k(sinx)^(k-1)sin(k-1)x-k(cosx)^(k-1)sin(k-1)x+2k(cos2x)^(k-1)sin2x

=k(sinx)^(k-1)sin(k-1)x-k(cosx)^(k-1)sin(k-1)x+k(cos2x)^(k-2)sin4x

可以检验当k=3时,f'(x)=0,此时必有f(x)为常数.