1.

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

Answer» `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`,
`:. f(x)_max = 25*25`
`=>(f(x)^(1/4))_max = (25*25)^(1/4)`
`=>(f(x)^(1/4))_max = (25)^(1/2) = 5.`
So, the maximum value of `f(x)^(1/4)` is `5`.


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