Differentiable Function In [0, 2], Find a and b. (where [.] Denotes The Greatest Integer Function).

If f (x) is differentiable function and (f (x). g(x)) is differentiable at x = a, then

Determine the value of ‘a’ if possible, so that the function is continuous

 for all real x and y. If f ’ (0) exists and equals to –1and f (0) = 1, find f ’(x).

Now if it is given that there exists a positive real δ, such that f (h) = h for 0 < h < δ then find f’(x) and hence f (x).

Let f  be an even function and f ’(0) exists, then find f’(0).

Let f (x) = xⁿ, n being non-negative integer. Then find the value of n for which the equality f’(a + b) = f ’(a) + f ’ (b) is valid for all, a, b > 0

Find the set of points where x² |x| is true thrice differentiable.

Find the number of points where f (x) = [sin x + cos x] (where [.] denotes greatest integral function), x Ïµ [0, 2π] is not continuous.

Discuss the continuity of the function .