JEE Advanced Level Test: Application of Derivative - JEE MCQ

The real number x (x > 0) when added to its reciprocal gives the minimum sum at x equals

If the function f (x) = 2x³ - 9ax² + 12a²x + 1, where a > 0 , attains its maximum and minimum at p and q respectively such that p² = q, then ‘a’ equals

If 2a + 3b + 6c = 0, then at least one root of the equation ax² + bx + c = 0 lies in the interval

A point on the parabola y² = 18x at which the ordinate increases at twice the rate of the abscissa, is

The normal to the curve  x = a (1 + cos θ) , y = a sin θ at 'θ' always passes through the fixed point

Angle between the tangents to the curve y = x² - 5x + 6 at the points (2, 0) and (3, 0) is

A function is matched below against an interval where it is supposed to be increasing, Which of the following pairs is incorrectly matched?

A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³/min. When the thickness of ice is 15 cm, then the rate at which the thickness of ice decreases, is

Area of the greatest rectangle that can be inscribed in the ellipse

Let f (x) be differentiable for all x. If f (1) =-2 and f '(x) > 2 for x ∈ (1, 6) then

The normal to the curve x = a (cos θ + θ sin θ), y = a (sin θ - θ cos θ) at any point 'θ' is such that

The function has a local minimum at 

A value of c for which the conclusion of Mean value Theorem holds for the function f (x) = log_e x on the interval [1, 3] is

The function f (x) =  tan^-1 (sin x cos x) is an increasing function in

Suppose the cubic x³ - px + q = 0 has three distinct real roots where p > 0 and q > 0. Then, which one of the following holds?

The shortest distance between the line y - x= 1 and the curve x =y² is

Given p (x) = x⁴ + ax³ + bx² + cx + d such that x = 0 is the only real root of p' (x) = 0 . If p (-1) < p (1) , then in the interval [-1,1]

The equation of the tangent to the curve  that is parallel to the x-axis is

Let f : R → R be defined by If f has a local minimum at x = - 1, then a possible value of k is

Let f : R → R be a continuous function defined by
Statement-1 :
Statement -2 :

The normal to the curve 

Consider the function, f (x) =| x - 2 | + | x - 5 | .x ∈ R
Statement -1 : f '(4) = 0
Statement -2 : f is continous in [2, 5], differentiable in (2, 5) and f (2) = f (5)

A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is

Let a, b ∈ R be such that the function f given by f (x) = In | x | +bx² + ax, x ≠ 0 has extreme values at x =-1 and x = 2.

Statement – 1: f has local maximum at x = - 1 and at x = 2.
Statement-2 : 

The real number k for which the equation, 2x³ + 3x + k = 0 has two distinct real roots in [0, 1]

The function f (x) = sin⁴ x+ cos⁴ x Increasing if 

dt then f decreasing in the interval