F (x) = SiNx (1 + cosx) maximum

F (x) = SiNx (1 + cosx) maximum

Derivation of composite function, (x) = (SiNx) ~ (1 + cosx) + SiNx (1 + cosx) ~ = cosx (1 + cosx) + SiNx (0-sinx) = cosx + (cosx) ^ 2 - (SiNx) ^ 2 = cosx + cos2x = cosx + 2 (cosx) ^ 2-1 (F ~ (x) denotes the derivative of F (x))
When f ~ (x) = 0, that is, cosx + 2 (cosx) ^ 2-1 = 0, the solution is cosx = 1 / 2, cosx = - 1
Since f (x) is a continuous function, the maximum value must appear in cosx = 1 / 2, cosx = - 1, cosx = 1
When cosx = - 1, cosx = 1, f (x) = 0
When cosx = 1 / 2, SiNx = √ 3 / 2, f (x) = 3, f (x) = 3 √ 3 / 4
When cosx = 1 / 2, SiNx = - √ 3 / 2, f (x) = 3, f (x) = - 3 √ 3 / 4
So the maximum value of F (x) = SiNx (1 + cosx) is 3 √ 3 / 4