Application of Derivatives questions
910 published questions with worked solutions.
- Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. If p(1) = 6 and...
- If f(x) = _0^x e^ t^2 (t - 2)(t - 3) ,dt for all x (0, ), then
- Let f : [ 1 2 , 1 ] R (the set of all real numbers) be a positive, non-constant and differentiable function such that f'(x) <...
- The number of points in (−∞, ∞), for which x^2 - x x - x = 0, is
- Let f(x) = x x, x > 0. Then for all natural numbers n, f'(x) vanishes at
- A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular...
- A line L : y = mx + 3 meets y-axis at E(0, 3) and the arc of the parabola y^2 = 16x, 0 y 6 at the point F(x_0, y_0). The tangent...
- Let a ∈ ℝ and let f : ℝ → ℝ be given by f(x) = x^5 - 5x + a. Then
- The slope of the tangent to the curve (y - x^5)^2 = x(1 + x^2)^2 at the point (1, 3) is
- List I P. Let y(x) = (3 ^ -1 x), x ∈ [−1, 1], x 3 2 . Then 1 y(x) (x^2 - 1) d^2y(x) dx^2 + x dy(x) dx equals Q. Let A_1, A_2, ,...
- A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of...
- Let f: R R be a continuous odd function, which vanishes exactly at one point and f(1)= 1 2 . Suppose that F(x)= _ -1 ^ x f(t) ,dt...
- Let f,g:[-1,2] R be continuous functions which are twice differentiable on the interval (-1,2). Let the values of f and g at the...
- The least value of R for which 4 x^2+ 1 x 1, for all x>0, is
- If f: R R is a twice differentiable function such that f''(x)>0 for all x R , and f ( 1 2 )= 1 2 , f(1)=1, then
- If f: R R is a differentiable function such that f'(x)>2f(x) for all x R , and f(0)=1, then
- For every twice differentiable function f: R [-2,2] with (f(0))^2+(f'(0))^2=85, which of the following statement(s) is (are) TRUE?
- Let denote a curve y=y(x) which is in the first quadrant and let the point (1,0) lie on it. Let the tangent to at a point P...
- Let f(x)= x x^2 , x>0. Let x_1<x_2<x_3< <x_n< be all the points of local maximum of f and y_1<y_2<y_3< <y_n< be all the points of...
- Let f: R R be given by f(x)=(x-1)(x-2)(x-5). Define F(x)= _0^xf(t) ,dt, x>0. Then which of the following options is/are correct?
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