Question

2.Find the intervals in which the function f(x) = tan−1
(sin + cos ), 0 < x < 2π is
strictly increasing and strictly decreasing . Also find their points of loal maxima
and local minima.
pls answer urgent with detailed step

Answer 1

Given : f(x)  = tan⁻¹(Sinx + Cosx)     0 < x < 2π  

To Find : intervals in which the function strictly increasing and strictly decreasing

points of loal maxima  and local minima.

Solution:

f(x)  = tan⁻¹(Sinx + Cosx)

f'(x) = ( 1/(1 +(Sinx + Cosx)²)) (cosx - Sinx)

=> f'(x) =  (cosx - Sinx)/(1 +(Sinx + Cosx)²)

(1 +(Sinx + Cosx)²) is always + ve

Hence

strictly increasing

f'(x)  >  0 if  cosx - Sinx  >  0  

=> Cosx > Sinx  

(0 , π /4) ∪ ( 5π /4 , 2π)

strictly Decreasing

f'(x)  <  0 if  cosx - Sinx  <  0  

=> Cosx <  Sinx  

( π /4 , 5π/4)

Local Maxima at     π /4

Local Minima  at     5π /4

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