f(x)=4sin(兀除以3加x)+sinx+2的最小值

Question

f(x)=4sin( 兀 加x)+sinx+2的最小值
除以3

Answer 1

 f(x) = 4sin(π/3 + x) + sinx + 2= 4sin(π/3)cosx + 4cos(π/3)sinx + sinx + 2= 4((√3)/2)cosx + 4(1/2)sinx + sinx + 2= (2√3)cosx + 2sinx + sinx + 2= (2√3)cosx + 3sinx + 2= √21(((2√3)/√21)cosx + (3/√21)sinx) + 2( 令 cosθ= (2√3)/√21則 sinθ= 3/√21 )= √21(cosθcosx + sinθsinx) + 2= √21cos(θ- x) + 2-1 ≤ cos(θ- x) ≤ 1- √21 + 2 ≤ √21cos(θ- x) + 2 ≤ √21 + 2- √21 + 2 ≤ f(x) ≤ √21 + 2故 f(x) 之最小值為 - √21 + 2