Applications of Derivatives (Monotonicity, Extrema): JEE Questions & Solutions

Let f: ℝ → ℝ be a twice differentiable function such that f''(x) > 0 for all x ∈ ℝ and f'(a-1) = 0, where a is a real number. Let g(x) =…

Let f(x)=x²⁰²⁵-x²⁰⁰⁰, x∈[0,1] and the minimum value of the function f(x) in the interval [0,1] be (80)⁸⁰(n)⁻⁸¹. Then n is equal to

Let (2α,α) be the largest interval in which the function f(t)=(|t+1|)/t², t<0 is strictly decreasing. Then the local maximum value of the…

Consider the following three statements for the function f: (0,∞) → ℝ defined by f(x) = | logₑ x | - |x - 1|: (I) f is differentiable at…

Let f be a differentiable function satisfying f(x)=1-2x+∫₀^x e^(x-t)f(t) dt, x∈ℝ, and let g(x)=∫₀^x {f(t)+2}¹⁵(t-4)⁶(t+12)¹⁷ dt. If p and q…

Let f(x) = ∫₀^x² (t² - 8t + 15)/e^t dt, x ∈ ℝ. Then the numbers of local maximum and local minimum points of f, respectively, are:

If the set of all values of a, for which the equation 5x³ - 15x - a = 0 has three distinct real roots, is the interval (α, β), then β - 2α…

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the…

Consider the region R = { (x, y): x ≤ y ≤ 9 - 11/3 x², x ≥ 0 }. The area of the largest rectangle of sides parallel to the coordinate axes…

Let (2, 3) be the largest open interval in which the function f(x) = 2 logₑ (x - 2) - x² + ax + 1 is strictly increasing, and (b, c) be the…

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

Let f: ℝ → ℝ be a function defined by f(x) = ||x+2| - 2|x||. If m is the number of points of local minima of f and n is the number of…

Let a > 0. If the function f(x) = 6x³ - 45ax² + 108a²x + 1 attains its local maximum and minimum values at the points x₁ and x₂…

Let x = -1 and x = 2 be the critical points of the function f(x) = x³ + ax² + b log|x| + 1, where x ≠ 0. Let m and M be the absolute…

If the range of the function f(x) = (5 - x)/(x² - 3x + 2), x ≠ 1, 2 is (-∞, α] ∪ [β, ∞), then α² + β² is equal to:

Let f: ℝ → ℝ be a polynomial function of degree four having extreme values at x = 4 and x = 5. If lim_(x → 0) f(x)/x² = 5, then f(2) is…

Let the function f(x) = x/3 + 3/x + 3, x ≠ 0, be strictly increasing in (-∞, α₁) ∪ (α₂, ∞) and strictly decreasing in (α₃, α₄) ∪ (α₄, α₅).…

For differentiable function f: (0, ∞) → R, if f(x) - f(y) ≥ logₑ (x/y) + x - y, find Σ f'(1/n²) from n=1 to 20.

The function f(x) = 2x + 3x^(2/3), x ∈ R, has:

The function f(x) = x/(x² - 6x - 16), x ∈ ℝ \ {-2, 8}, has:

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 (x, y) and…

Let f(x) = (x + 3)² (x - 2)³, x ∈ [-4, 4]. If M and m are the maximum and minimum values of f in [-4, 4], then the value of M - m is:

The number of critical points of f(x) = (x - 2)^(2/3) (2x + 1) is:

The number of points of local maxima of f(x) = 4 cos³(x) + 3√3 cos²(x) - 10 in the interval (0, 2π) is:

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space S = {x ∈ ℤ: (66 - x)x ≥ 5/9M}…

Let x = 2 be a local minima of the function f(x) = 2x⁴ - 18x² + 8x + 12, x ∈ (-4, 4). If M is the local maximum value of the function f(x)…

Let the function f(x) = 2x³ + (2p - 7)x² + 3(2p - 9)x - 6 have a maxima for some value of x < 0 and a minima for some value of x > 0. Then,…

If the equation of the normal to the curve y = (x - a)/(x + b)(x - 2) at the point (1, -3) is x - 4y = 13, then the value of a + b is:

The number of points on the curve y = 54x⁵ - 135x⁴ - 70x³ + 180x² + 210x at which the normal lines are parallel to x + 90y + 2 = 0 is:

If the functions f(x) = x³/3 + 2bx + ax²/2 and g(x) = x³/3 + ax + bx², a ≠ 2b have a common extreme point, then a + 2b + 7 is equal to:

A wire of length 20 m is to be cut into two pieces. A piece of length ℓ₁ is bent to make a square of area A₁, and the other piece of length…

Let y = f(x) = sin³ ( π/3 cos ( π/3√2 ( -4x³ + 5x² + 1)^(3/2))) Then, at x = 1,

The equation e^4x + 8e^3x + 13e^2x - 8e^x + 1 = 0, x ∈ ℝ has:

The absolute minimum value of the function f(x) = |x² - x + 1| + ⌊ x² - x + 1 ⌋, where [t] denotes the greatest integer function, in the…

The sum of the absolute maximum and minimum values of the function f(x) = |x² - 5x + 6| - 3x + 2 in the interval [-1, 3] is equal to:

Let a curve y = f(x), x ∈ (0, ∞) pass through the points P(1, 3/2) and Q(a, 1/2). If the tangent at any point R (b, f(b)) to the given…

The number of points, where the curve y = x⁵ - 20x³ + 50x + 2 crosses the x-axis is:

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to…

If f(x) = [(tan 1^°)x + logₑ(123)]/[x logₑ(1234) - (tan 1^°)], x > 0, then the least value of f(f(x)) + f(f(4/x)) is:

Let g(x) = f(x) + f(1 - x) and f''(x) > 0 for all x ∈ (0, 1). If g is decreasing in the interval (0, α) and increasing in the interval (α,…

Let the quadratic curve passing through the point (-1, 0) and touching the line y = x at (1, 1) be y = f(x). Then the x-intercept of the…

Let a, b, c, and d be positive real numbers such that a + b + c + d = 11. If the maximum value of a⁵ b³ c² d is 3750β, then the value of β…

If the local maximum value of the function f(x) = ( √3e/(2 sin x))^(sin² x), x ∈ ( 0, π/2) is k/e, then ( k/e)⁸ + k⁸/e⁵ + k⁸ is equal to:

Find the maximum value of the function max_(0 ≤ x ≤ π){ x - 2sin xcos x + 1/3sin 3x }

Consider the triangles with vertices A(2,1), B(0,0) and C(t,4), where t∈[0,4]. If the maximum and the minimum perimeters of such triangles…