Find the points of extrema of the function f(x)=2 sec x+3 cosec x

Explanation

f(x)=2 sec x + 3 cosec x
f(x)= 2 sec `xx` tan x-3 cosec x `xx` cot x
Let f(x) =0
`2(sinx)/(cos^(2)x)=3(cosx)/(sin^(2)x)`
`tan^(3)x=3//2`
`x=npi+tan^(-1)(3/2)^(1//3),n` in Z.
(i) if x in `(0,pi//2)`, then x = `tan^(-1)(3/2)^(1//3)`
Now lim (2 sec x + 3 cosecs) `rarr` 00 and
lim (2 sec x + 3cos ecs) `rarr`00
`xrarr(pi)/(2)`
`therefore x=tan^(-1)(3/2)^(1//3)` is point of minima.
(ii) if x in `(pi,3pi//2)`,then x = `pi+tan^(-1)(3/2)^(1//3)`
Now lim(2sec x+3cosec) `rarr`-00
`therefore x=pi+tan^(-1)(3/2)^(1//3)` is point of maxima.