Let `f(x)=sinx+cosx+tanx+sin^(-1)x+cos^(-1)x+tan^(-1)xdot` Then find the maximum and minimum values of `f(x)dot`

Explanation

`f(x) = sin x + cos x + tan x + siN^(_1) x + cos^(-1) x + tan^(-1)`
`f(x) = sin x + cos y + tan x + (pi)/(2) + tan^(-1) x`
Clearly, domain of `f` is `[-1, 1]`
`f(x) = cos x - sin x + sec^(2) x + 0 + (1)/(1 + x^(2))`
Now, `sec^(2) x ge and (1)/(1 + x^(2)) in [(1)/(2), 1]`
Hence, `f'(x) gt 0`
`rArr f` is increasing
`rArr` Range is `[f(-1), f(1)]`
Now, `f(-1) = - sin 1 + cos 1 - tan 1 + (pi)/(2) - (pi)/(4) = (pi)/(4) + cos 1 - sin 1 - tan 1`
And `f(1) = sin 1 + cos 1 + tan 1 + (pi)/(2) + (pi)/(4)`
`= (3pi)/(4) + cos 1 + sin 1 + tan 1`
Thus, least value is `(pi)/(4) + cos 1 - sin 1 - tan 1` and greatest value is `(3pi)/(4) + cos 1 + sin 1 + tan 1`