Question

What is the maximum and minimum value of cos(cosx)

Answer 1

For such questions it only needs some logic to solve. We don't need to do maxima and minima from differentiation
The maximum and minimum value of cos(x) is 1 and -1.
Hence cos(cosx) = at , x = 90 degree is cos0 = 1 (maximum value )
And cos(cosx) = at x = 0 is cos(1) , it is minimum value .

Here the cos function decreases from 0 to 90 degree, so it has smooth relations between angles .
Hence if one cos increases then other cos will decreases simultaneously.
That's it.