Continuity and Differentiability questions
732 published questions with worked solutions.
- f, g : ℝ ℝ be two real valued functions defined as f(x) = −|x + 3| for x < 0 and e^ x for x 0, and g(x) = x^ 2 + k_ 1 x for x < 0...
- Let f : R R be a function such that f(x + y) = f(x) + f(y), x, y R . If f(x) is differentiable at x = 0, then
- If f(x) = cases -x - 2 , & x - 2 - x, & - 2 < x 0 x - 1, & 0 < x 1 x, & x > 1, cases then
- Let f(x) = cases x^2 | x |, & x 0 0, & x = 0 cases , x R , then f is
- For every integer n, let a_n and b_n be real numbers. Let function f : R R be given by f(x) = cases a_n + x, & for x [2n, 2n + 1]...
- For every pair of continuous functions f, g : [0, 1] → ℝ such that f(x) : x [0, 1] = g(x) : x [0, 1] , the correct statement(s)...
- Let f : [a, b] → [1, ∞) be a continuous function and let g : ℝ → ℝ be defined as g(x) = cases 0 & if x < a, _a^x f(t) ,dt & if a...
- Let f : ℝ → ℝ and g : ℝ → ℝ be respectively given by f(x) = |x| + 1 and g(x) = x^2 + 1. Define h : ℝ → ℝ by h(x) = cases f(x),...
- Let g: R R be a differentiable function with g(0)=0, g'(0)=0 and g'(1) 0. Let f(x)= cases x |x| g(x), & x 0 0, & x=0 cases and...
- Let a,b R and f: R R be defined by f(x)=a (|x^3-x|)+b|x| (|x^3+x|). Then f is
- Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f(x)=x ( (x+[x]))...
- Let f: R (0,1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the...
- Let f:(0, ) R be a twice differentiable function such that _ t x f(x) t-f(t) x t-x = ^2x for all x (0, ). If f ( 6 )=- 12 , then...
- Let f(x) = cases x^2 - 1, & 0 < x < 2 2x + 3, & 2 x < 3 cases , the quadratic equation whose roots are _ x 2^- f(x) and _ x 2^+...
- Consider the function f(x) = x. Then,
- Let f(x) = cases k x - 2x , & when x 2 3, & when x = 2 cases If _ x 2 f(x) = f ( 2 ), then k is equal to
- If f(x) = cases mx^2 + n, & x < 0 nx + m, & 0 x 1 nx^3 + m, & x > 1 cases , then
- Suppose f(x) = cases a + bx, & x < 1 4, & x = 1 b - ax, & x > 1 cases and if _ x 1 f(x) = f(1), then
- The function f defined by f(x) = 3^x + 3^ -x - 2 x^2 is continuous, the value of f(0) is
- If f(x) = (3^x - 1)^2 x (1 + x) is continuous, the value of f(0) is
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