Prove that `f(x) = [x], 0 lt x lt 3` is not differentiable at `x = 1` but `x = 2`.

Prove that f(x) = |x| is continous but not differentiable at x = 0

Prove that the greatest integer function defined by f(x) = [x], 0

If a function f(x) is defined as f(x) = {{:(-x",",x lt 0),(x^(2)",",0 le x le 1),(x^(2)-x + 1",",x gt 1):} then a. f(x) is differentiable at x = 0 and x = 1 b. f(x) is differentiable at x = 0 but not at x = 1 c. f(x) is not differentiable at x = 1 but not at x = 0 d. f(x) is not differentiable at x = 0 and x = 1

Prove that the function f given by f(x) = |x-1|, x in R, x=1 is not differentiable at x = 1.

If f(x) = 0 for x lt 0 and f(x) is differentiable at x = 0, then for x gt 0, f(x) may be

Lt f(x) = |cos x|, then

If f(x) = 3(2x + 3)^(2//3) + 2x + 3 , then: (a) f(x) is continuous but not differentiable at x = - (3)/(2) (b) f(x) is differentiable at x = 0 (c) f(x) is continuous at x = 0 (d) f(x) is differentiable but not continuous at x = - (3)/(2)

If 0 lt x lt pi and cos x + sin x = 1/2 , then tan x is

If f (x)= [{:((sin [x^(2)]pi)/(x ^(2)-3x+8)+ax ^(3)+b,,, 0 le x le 1),( 2 cos pix + tan ^(-1)x ,,, 1 lt x le 2):} is differentiable in [0,2] then: ([.] denotes greatest integer function)