The maximum value of F (x) = cos (2arccosx) + 4sin (arcsinx / 2)

f(x)=cos(2arccosx)+4sin[arcsin(x/2)]
= 2[cos(arccosx)]^2 - 1 + 4 * (x/2)
= 2x^2 -1 + 2x
= 2x^2 + 2x - 1
= 2(x^2 + x) - 1
= 2(x^2 + x + 1/4 - 1/4) - 1
= 2[(x + 1/2)^2 - 1/4] - 1
= 2(x + 1/2)^2 - 3/2
The domain of X is [- 1,1]
therefore
When x = - 1 / 2, the minimum value of F (x) is - 3 / 2
When x = 1, the maximum value of F (x) is 3

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