Maths Continuity and Differentiability

The function f(x)   $=\frac{4x^3-3x^2}{6}-2sinx+\left(2x-1\right)cosx:$

Let f be differentiable on R and h(x) = f(x) − (f(x))² + (f(x))³ + 3x + 2 cos x, then

Let f: (-1, 1) → R be a differentiable function with f(0) = 1 and f'(0) = -1. If g(x) = (f(nf(x) – n))^n, where n is a natural number and g'(0) = 4, then n equals

Let f : [0,∞) → be a function defined by f(x) = { max{sint: 0≤ t ≤x}, 0≤x≤π, 2 + cos x, x > π
Then which of the following is true?

Let S be the set of points where the function, f(x) = |2 − |x − 3, x ∈ R, is not differentiable. Then Σ [x∈S] f(f(x)) is equal to

Let the functions f: R → R and g: R → R be defined as :
f(x) = {x+2, x<0; x, x0} and g(x) = {x, x<1; 3x-2, x1}
Then, the number of points in R where (fog)(x) is NOT differentiable is equal to :

Let f: R→R and g: R → R be defined as f(x) = {x+a, if x<0; |x-1|, if x0} and g(x) = {x+1, if x<0; (x-1)+b, if x0}, where a, b are non-negative real numbers. If (gof)(x) is continuous for all x R, then a + b is equal to.

Let f:S → S where S = (0,∞) be a twice differentiable function such that f(x+1) = xf(x). If g: S → R be defined as g(x) = logₑf(x), then the value of |g''(5)-g''(1)| is equal to :

Let α ∈ R be such that the function f(x) = { [cos⁻¹(1-{x}²)sin⁻¹(1-{x})] / ({x}-{x}³), x≠0; α, x=0 } is continuous at x=0, where {x} = x-[x], [x] is the greatest integer less than or equal to x. Then :

Let f be a real valued function, defined on R - {-1, 1} and given by f(x) = 3logₑ|(x-1)/(x+1)| - 2/(x-1). Then in which of the following intervals, function f(x) is increasing?