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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).

```
11 years ago

```							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 135o
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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Naga Ramesh
IIT Kgp - 2005 batch

```
11 years ago
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