If `f(x)=2(7cosx+24sinx)(7sinx-24cosx),` for every `x in R ,` then maximum value of `f(x)^(1/4)` is_____

Explanation

`f(x) = 2(7cosx+24sinx)(7sinx-24cosx)`

`=>f(x) = 2*25*25(7/25cosx+24/25sinx)(7/25sinx-24/25cosx)`

Let `cos alpha = 7/25`,

Then, `sinalpha = sqrt(1-(7/25)^2) = sqrt(576/625) = 24/25`

`:. f(x) = 2*25*25(cosalphacosx+sinalphasinx)(cosalphasinx-cosalphacosx)`

`=>f(x) = 25*25*2(cos(x-alpha)sin(x-alpha))`

`=>f(x) = 25*25*sin(2(x-alpha))`

As maximum value of `sin(2(x-alpha))` is `1`,

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