试求正整数k,使f(x)=sinkx*sin^k(x)+coskx*cos^k(x)—cos^k(2x)的值不依赖于x

题目

试求正整数k,使f(x)=sinkx*sin^k(x)+coskx*cos^k(x)—cos^k(2x)的值不依赖于x

答案

代入x = 0可得f(0) = 0.代入x = π/k,可得f(π/k) = -cos(π/k)^k-cos(2π/k)^k.若f(x)不依赖于x,则有-cos(π/k)^k-cos(2π/k)^k = f(π/k) = f(0) = 0.若k为偶数,有-cos(π/k)^k ≤ 0且-cos(2π/k)^k ≤ 0.而由cos(...