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) = f( ^2 x…

Let f(x)=x^2025-x^2000, 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 (2,) be the largest interval in which the function f(t)= |t+1|t^2, 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) = | e x | - |x - 1|: (I) f is differentiable at all x 0.…

Let f be a differentiable function satisfying f(x)=1-2x+ 0^x (t-x)f(t) dt, x R, and let g(x)= 0^x f(t)+2 ^5(t-4)^6(t+12)^7 dt. If p and q…

Let f(x) = 0^x^2 t^2 - 8t + 15e^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 5x^3 - 15x - a = 0 has three distinct real roots, is the interval (,), then - 2 is…

Consider the region R = (x, y): x y 9 - 113 x^2, x 0. The area of the largest rectangle of sides parallel to the coordinate axes and…

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

If the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1, where a 0, attains its local maximum and minimum at p and q, respectively, such that p^2…

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

MATHEMATICS SECTION-A Let a 0. If the function f(x) = 6x^3 - 45ax^2 + 108a^2x + 1 attains its local maximum and minimum values at the…

Let x = -1 and x = 2 be the critical points of the function f(x) = x^3 + ax^2 + b |x| + 1, where x 0. Let m and M be the absolute minimum…

If the range of the function f(x) = 5 - xx^2 - 3x + 2, x 1, 2 is (-, ] [,), then ^2 + ^2 is equal to:

Let f: R R be a polynomial function of degree four having extreme values at x = 4 and x = 5. If x 0 f(x)x^2 = 5, then f(2) is equal to:

Let the function f(x) = x3 + 3x + 3, x 0, be strictly increasing in (-, 1) ( 2,) and strictly decreasing in ( 3, 4) ( 5, s). Then i=1^5 i^2…

For differentiable function f: (0,) R, if f(x) - f(y) e ( xy) + x - y, find f' ( 1n^2) from n=1 to 20.

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

The function f(x) = xx^2 - 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 = -2x^2 + 54 at points (x, y) and…

Let f(x) = (x + 3)(x - 2)^2(x + 1), 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 ^3(x) + 3 3 ^2(x) - 10 in the interval (0, 2) 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 59M and…

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

Let the function f(x) = 2x^3 + (2p - 7)x^2 + 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 = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x at which the normal lines are parallel to x + 90y + 2 = 0 is:

If the functions f(x) = x^33 + 2bx + ax2 and g(x) = x^33 + ax + bx^2, 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 1 is bent to make a square of area A1, and the other piece of length…

Let y = f(x) = ^3 ( 3 ( 3 2 ( -4x^3 + 5x^2 + 1)^ 32)) Then, at x = 1,

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

The absolute minimum value of the function f(x) = |x^2 - x + 1| + x^2 - 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) = |x^2 - 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, 32) and Q(a, 12). If the tangent at any point R (b, f(b)) to the given curve cuts…

The number of points, where the curve y = x^5 - 20x^3 + 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) = [( 1^)x + e(123)][x e(1234) - ( 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^(n)(x) 0, x (0, 1). If g is decreasing in the interval (0,) and increasing in the interval (, 1), then…

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 a^5 b^3 c^2 d is 3750, then the value of…

If the local maximum value of the function f(x) = ( 3e2 x)^ ^2 x, x ( 0, 2) is ke, then ( ke)^8 + k^8e^5 + k^8 is equal to:

Find the maximum value of the function

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…