Continuity and Differentiability questions
732 published questions with worked solutions.
- Let f : [1, 3] → R be a function satisfying x [x] f(x) 6 - x , for all x ≠ 2 and f(2) = 1, where R is the set of all real numbers...
- If f(x) = a| x| + be^ |x| + c|x|^3, where a, b, c ∈ R, is differentiable at x = 0, then
- If f : R R is a function defined by f(x) = [x] ( 2x - 1 2 ) , where [x] denotes the greatest integer function, then f is
- Consider the function f(x) = |x − 2| + |x − 5|, x ∈ R. Statement 1: f′(4) = 0 Statement 2: f is continuous in [2, 5],...
- Statement-1: The equation x log x = 2 − x is satisfied by at least one value of x lying between 1 and 2. Statement-2: The...
- Let f(x) = −1 + |x − 2|, and g(x) = 1 − |x|; then the set of all points where fog is discontinuous is :
- Let f be a composite function of x defined by f(u) = 1 u^2 + u - 2 , u(x) = 1 x - 1 . Then the number of points x where f is...
- Consider the function : f(x) = [x] + |1 − x|, −1 ≤ x ≤ 3 where [x] is the greatest integer function. Statement 1: f is not...
- Let f, g : R R be two functions defined by f(x) = cases x ( 1x ), & x 0 0, & x = 0 cases and g(x) = xf(x). Statement I: f is a...
- If the function f(x) = cases 2 + x - 1 ( - x)^2 , & x , & x = cases is continuous at x = , then k equals
- Let f : R R be a function such that |f(x)| x^2, for all x R. Then, at x = 0, f is
- Let ℝ denote the set of all real numbers. Let f : ℝ → ℝ be an arbitrary function and let g : ℝ → ℝ be the function defined by...
- Consider the function f : (- 2 , 2 ) (- , ) defined by f(x) = (|x| + |x - 1|) x + [x x], where [x sin x] is the greatest integer...
- For a real number , let [ ] denote the greatest integer less than or equal to . For a finite set S, let |S| denote the number of...
- If y = | x| + | x| then dy dx at x = 2 3 is
- If f(x) = 2 [x] - x^3 , 1<x<2, and [x] = the greatest integer x, then f' ! ( [3] 2 ) is equal to
- If P(x) is a polynomial such that P(x^2+1) = P(x) ^2 + 1 and P(0) = 0 then P'(0) is equal to
- If f(x) = x^3 + x^2f'(1) + xf''(2) + f'''(3) for all x R then
- Let y = | ( 4 -x ) |. Then dy dx at x= 4
- Let y = |x| + |x-2|. Then dy dx at x=2
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