Show that the extremum value of the function f(x) = 1 / (sinx+4) - 1 / (cos x -4), where x belongs to R is 2√2 / (4 √2 + 1).

f '(x) = -cos x/(4+sin x)² -sin x/(-4 +cos x)²

sin x.(4+sin x)² +cos x.(-4 +cos x)² = 0

(cos x + sin x)(8sin x -8 cos x + cos x.sin x +17) = 0

as (8sin x -8 cos x + cos x.sin x +17) cant be zero since  -1<sinx<1 and -1<cosx<1

so, (cos x + sin x) = 0

which mean cos x = -sin x

x = 2nPi + 3Pi/4

one possible value is 135^o

on substitution, we get

f(x) = 1/(4+sin x) -1/(-4 +cos x)

f(x) = 2/(4+sin x)

put sin x = 2^-1/2

which gives extreme value of f(x) as 2√2 / (4 √2 + 1)

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