Application of Derivatives questions
910 published questions with worked solutions.
- The tangents to the curve y = (x - 2)^2 - 1 at its points of intersection with the line x − y = 3, intersect at the point:
- Let f(x) be a polynomial of degree 5 such that x= 1 are its critical points. If _ x 0 (2+ f(x) x^3 )=4, then which one of the...
- The value of c in the Lagrange's mean value theorem for the function f(x)=x^3-4x^2+8x+11, when x [0,1] is:
- Let a function f:[0,5] R be continuous, f(1) = 3 and F be defined as F(x)= _1^x t^2g(t) ,dt, where g(t)= _1^t f(u) ,du. Then for...
- Let f be any function continuous on [a, b] and twice differentiable on (a, b). If for all x (a, b), f'(x) > 0 and f''(x) < 0,...
- The number of real roots of the equation e^ 4x +e^ 3x -4e^ 2x +e^x+1=0 is
- If the surface area of a cube is increasing at a rate of 3.6 cm ^2/sec, retaining its shape, then the rate of change of its...
- Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If T= x R|f(x)=f(0) , then the sum of squares of...
- If the tangent to the curve, y=e^x at a point (c,e^c) and the normal to the parabola, y^2=4x at the point (1, 2) intersect at the...
- If 5f(x) + 4f ( 1x ) = x^2 - 2, ∀ x ≠ 0 and y = 9x²f(x), then y is strictly increasing in:
- Let f(x) = 3 x - 2 + 4 - x be a real valued function. If α and β are respectively the minimum and the maximum values of f, then...
- Let f : R → R be a thrice differentiable function such that f(0) = 0, f(1) = 1, f(2) = –1, f(3) = 2 and f(4) = –2. Then, the...
- Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on...
- For the function f(x) = x + 3x - 2 (x^2 + x), where x ∈ [0, 2 ]. Consider the following two statements: (I) f is increasing in...
- Let the maximum and minimum values of ( 8x - x^2 - 12 - 4 )^2 + (x - 7)^2, x ∈ R be M and m respectively. Then M² − m² is equal...
- The interval in which the function f(x) = xˣ, x > 0, is strictly increasing is
- Let the function, f : [–7, 0] → R be continuous on [–7, 0] and differentiable on (–7, 0). If f(–7) = – 3 and f'(x) 2, for all x...
- Let the line y = mx and the ellipse 2x^2+y^2=1 intersect at a point P in the first quadrant. If the normal to this ellipse at P...
- If c is a point at which Rolle's theorem holds for the function, f(x)= _e ( x^2+ 7x ) in the interval [3, 4], where R, then...
- Let f:R R be such that for all x R (2^ 1+x +2^ 1-x ), f(x) and (3^x+3^ -x ) are in A.P., then the minimum value of f(x) is -
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