Continuity and Differentiability questions
732 published questions with worked solutions.
- If the function f(x) = 3x + x - 3x x^3 , x ∈ R, is continuous at x = 0, then f(0) is equal to:
- Let f : [−1, 2] → R be given by f(x) = 2x² + x + [x²] − [x], where [t] denotes the greatest integer less than or equal to t. The...
- If f(x) = cases x^3 ( 1x ), & x 0 0, & x = 0 cases , then
- Let S be the set of points where the function, f(x) = |2– | x – 3||, x R, is not differentiable. Then _ x S f(f(x)) is equal to ….
- Let f(x)=x ^ -1 (- |x|), x [- 2 , 2 ], then which of the following is true ?
- Let [t] denotes the greatest less than or equal to t. Let f : [0, ∞) → R be a function defined by f(x) = [ x2 + 3 ] - [ x]. Let S...
- If a function f(x) defined by f(x)= cases ae^x+be^ -x , & -1 x<1 cx^2, & 1 x 3 ax^2+2cx, & 3<x 4 cases be continuous for some a,...
- For a, b > 0, let f(x) = cases ((a + 1)x) + b x x , & x < 0 3, & x = 0 ax + b^2x^2 - ax b a ,x x , & x > 0 cases be a continuous...
- The function f(x)= cases 4 + ^ -1 x, & |x| 1 1 2 (|x|-1), & |x|>1 cases is
- Let f:(0, ) (0, ) be a differentiable function such that f(1) = e and _ t x t^2f^2(x)-x^2f^2(t) t-x =0. If f(x) = 1, then x is...
- If the function f(x)= cases k_1(x- )^2-1, & x _2 x, & x> cases is twice differentiable, then the ordered pair (k_1,k_2) is equal...
- Let f(x)=x [ x 2 ], for – 10 < x < 10, where [t] denotes the greatest integer function. Then the number of points of...
- Let f : R → R be defined as f(x)= cases x^5 ( 1 x )+5x^2, & x<0 0, & x=0 ^5 ( 1 x )+ x^2, & x>0 cases . The value of for which f...
- Let f : R → R be a function defined by f(x) = max x, x^2 . Let S denote the set of all points in R, where f is not...
- The number of points in the interval [2, 4], at which the function f(x) = [x^2 - x - 12 ], where [·] denotes the greatest integer...
- The number of points, at which the function f(x) = 6x, 2+3x^2 + |x-1| | |x^2- 14 | |, x ∈ (−π, π), is not differentiable, is...
- Let f(x)= cases e^ x-1 , & x<0 x^2-5x+6, & x 0 cases and g(x)=f(|x|)+|f(x)|. If the number of points where g is not continuous...
- Let f(x)= cases x^3+8, & x<0 x^2-4, & x 0 cases and g(x)= cases (x-8)^ 1/3 , & x<0 (x+4)^ 1/2 , & x 0 cases . Then the number of...
- For the function f(x) = e^ sin|x| − |x|, x ∈ ℝ, consider the following statements: Statement I: f is differentiable for all x ∈...
- Let f(x)= cases 13, & x /2 [2mm] b(1- x) ( -2x)^2 , & x> /2 cases . If f is continuous at x = π/2, then the value of _0^ 3b-6...
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