Show that f(x) = x + cos x – a is an increasing function on R for all values of a ?

Find the value(s) of a for which f(x) = – ax is an increasing function on R ?

Find the values of b for which the function f(x) = sin x – bx + c is a decreasing function on R ?

Let F defined on [0, 1] be twice differentiable such that | f”(x) ≤ 1 for all x ϵ [0, 1]. If f(0) = f(1), then show that |f’(x) | < 1 for all x ϵ [0, 1] ?

Find the intervals in which f(x) is increasing or decreasing :

i. f(x) = x |x|, x ϵ R

ii. f(x) = sin x + |sin x|, 0 < x ≤ 2 π

iii. f(x) = sin x (1 + cos x), 0 < x < π/2