Application of Derivatives questions
910 published questions with worked solutions.
- Let f : ℝ → ℝ be a differentiable function such that f ( x+y 3 )= f(x)+f(y) 3 for all x,y∈ℝ, and f′(0)=3. Then the minimum value...
- For the functions f(θ)=α tan²θ+β cot²θ and g(θ)=α sin²θ+β cos²θ, α>β>0, let _ 0< < /2 f( )= _ 0< < g( ). If the first term of a...
- Let the set of all values of p, for which f(x) = (p² − 6p + 8)(sin²2x − cos²2x) + 2(2 − p)x + 7 does not have any critical point,...
- Let the set of all positive values of λ, for which the point of local minimum of the function (1 + x(λ² − x²)) satisfies x^2 + x...
- Consider the function f : (0, 2) → R defined by f(x) = x2 + 2x and the function g(x) defined by g(x) = cases f(t) , & 0 < t x and...
- Let g(x) = 3f ( x3 ) + f(3 - x) and f″(x) > 0 for all x ∈ (0, 3). If g is decreasing in (0, α) and increasing in (α, 3), then 8α...
- If the shortest distance of the parabola y² = 4x from the centre of the circle x² + y² – 4x – 16y + 64 = 0 is d then, d² is
- Let for a differentiable function f : (0, ∞) → R, f(x) - f(y) _e ( xy ) + x - y, ∀ x, y ∈ (0, ∞). Then _ n=1 ^ 20 f' ( 1 n^2 ) is...
- The function f(x) = x x^2 - 6x - 16 , x ∈ ℝ − −2, 8
- The function f(x) = 2x + 3x^ 2/3 , x ∈ ℝ, has
- Let f(x) = 2ˣ – x², x ∈ ℝ. If m and n are respectively the number of points at which the curves y = f(x) and y = f ′(x) intersect...
- Let y = f(x) be a thrice differentiable function in (–5, 5). Let the tangents to the curve y = f(x) at (1, f(1)) and (3, f(3))...
- Let f(x) = (x + 3)²(x – 2)³, x ∈ [–4, 4]. If M and m are the maximum and minimum values of f, respectively in [–4, 4], then the...
- Let g : ℝ → ℝ be a non constant twice differentiable function such that g′(1/2) = g′(3/2). If a real valued function f is defined...
- The maximum area of a triangle whose one vertex is at (0, 0) and the other two vertices lie on the curve y = –2x² + 54 at points...
- If the function f : (−∞, −1] → (a, b] defined by f(x) = e^ x^3 - 3x + 1 is one-one and onto, then the distance of the point P(2b...
- Let a variable line passing through the centre of the circle x² + y² − 16x − 4y = 0, meet the positive coordinate axes at the...
- Let S = (−1, ∞) and f : S → ℝ be define as f(x) = _ -1 ^x(e^t - 1)^ 11 (2t - 1)^5(t - 2)^7(t - 3)^ 12 (2t - 10)^ 61 dt. Let p =...
- The function f(x)=x^3-6x^2+ax+b is such that f(2) = f(4) = 0. Consider two statements. (S_1) there exists x_1, x_2 (2,4),...
- Let be the acute angle between the tangents to the ellipse x^2 9 + y^2 1 =1 and the circle x^2+y^2=3 at their point of...
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