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JEE Main Application of Derivatives questions

55 Mathematics questions on Application of Derivatives as asked in JEE Main, newest first. 34 come with a worked solution.

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

    2023

  2. 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. Then, the set of all values of p is

    2023

  3. Let a tangent to the curve 9x^2 + 16y^2 = 144 intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is __________

    2023 · with solution

  4. If the tangent to the curve y = x/(x^2 - 3), x ∈ R, (x ≠ ±√(3)), at a point (α, β) ≠ (0, 0) on it is parallel to the line 2x + 6y - 11 = 0, then

    2019

  5. Let f(x) = 5 - |x - 2| and g(x) = |x + 1|, x ∈ R. If f(x) attains maximum value at α and g(x) attains minimum value of β, then lim_x → -αβ ((x-1)(x^2 - 5x + 6))/(x^2 - 6x + 8) is equal to

    2019 · with solution

  6. A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the…

    2019

  7. Let S be the set of all values of x for which the tangent to the curve y = f(x) = x^3 - x^2 - 2x at (x, y) is parallel to the line segment joining the points (1, f(1)) and (-1, f(-1)), then S is equal to

    2019 · with solution

  8. If the tangent to the curve, y = x^3 + ax - b at the point (1, -5) is perpendicular to the line, -x + y + 4 = 0, then which one of the following points lies on the curve?

    2019 · with solution

  9. If m is the minimum value of k for which the function f(x) = x√(kx - x^2) is increasing in the interval [0, 3] and M is the maximum value of f in the interval [0, 3] when k = m, then the ordered pair (m, M) is equal to

    2019 · with solution

  10. If f(x) is a non-zero polynomial of degree four, having local extreme points at x = -1, 0, 1, then the set S = \x ∈ R : f(x) = f(0)\ contains exactly

    2019 · with solution

  11. The tangent to the curve y = x^2 - 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point

    2019 · with solution

  12. Let f:[0,2] → R be a twice differentiable function such that f''(x) > 0, for all x ∈ (0,2). If φ(x) = f(x) + f(2-x), then φ is

    2019 · with solution

  13. A helicopter is flying along the curve given by y - x^3/2 = 7, (x ≥ 0). A soldier positioned at the point (1/2, 7 ) wants to shoot down the helicopter when it is nearest to him. Then, this nearest distance is

    2019 · with solution

  14. If the function f given by f(x) = x^3 - 3(a-2)x^2 + 3ax + 7, for some a ∈ R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, frac(x-1)^-14(x-1)^2 = 0 (x ≠ 1) is

    2019 · with solution

  15. The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is

    2019 · with solution

  16. Let f(x) = dfracx√(a^2 + x^2) - dfracd - x√(b^2 + (d-x)^2), x ∈ R, where a, b and d are non-zero real constants. Then,

    2019

  17. If S_1 and S_2 are respectively the sets of local minimum and local maximum points of the function, f(x) = 9x^4 + 12x^3 - 36x^2 + 25, x ∈ R, then

    2019

  18. If θ denotes the acute angle between the curves, y = 10 - x^2 and y = 2 + x^2 at a point of their intersection, then |tanθ| is equal to

    2019

  19. The shortest distance between the line y = x and the curve y^2 = x - 2 is

    2019 · with solution

  20. The maximum area (in sq. units) of a rectangle having its base on the X-axis and its other two vertices on the parabola, y = 12 - x^2 such that the rectangle lies inside the parabola, is

    2019 · with solution

  21. The maximum value of the function f(x) = 3x^3 - 18x^2 + 27x - 40 on the set S = \x ∈ R : x^2 + 30 ≤ 11x\ is

    2019 · with solution

  22. Let A(4, -4) and B(9, 6) be points on the parabola, y^2 = 4x. Let C be chosen on an arc AOB of the parabola, where O is the origin, such that the area of triangle ACB is maximum. Then, the area (in sq. units) of…

    2019 · with solution

  23. The maximum volume (in cu.m) of the right circular cone having slant height 3m is

    2019 · with solution

  24. The tangent to the curve, y = xe^x^2 passing through the point (1, e) also passes through the point

    2019 · with solution

  25. The shortest distance between the point (3/2, 0 ) and the curve y = √(x), (x > 0), is

    2019 · with solution

  26. If the curves y^2 = 6x, 9x^2 + by^2 = 16 intersect each other at right angles, then the value of b is

    2018 · with solution

  27. If f: R → R is a twice differentiable function such that f''(x) > 0 for all x ∈ R, and f (1/2 ) = 1/2, f(1) = 1, then

    2017 · with solution

  28. The normal to the curve y(x-2)(x-3) = x + 6 at the point, where the curve intersects the Y-axis passes through the point

    2017

  29. Consider f(x) = tan^-1 (√((1+sin x)/(1-sin x)) ), x ∈ (0, (π)/2 ). A normal to y = f(x) at x = (π)/6 also passes through the point

    2016

  30. A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum,…

    2016 · with solution

  31. The least value of α ∈ R for which 4α x^2 + 1/x ≥ 1, for all x > 0, is

    2016 · with solution

  32. The normal to the curve x^2 + 2xy - 3y^2 = 0 at (1,1)

    2015 · with solution

  33. If x = -1 and x = 2 are extreme points of f(x) = α log|x| + β x^2 + x, then

    2014 · with solution

  34. The number of points in (-∞, ∞) for which x^2 - xsin x - cos x = 0, is

    2013

  35. Let f, g and h be real-valued functions defined on the interval [0, 1] by f(x) = e^x^2 + e^-x^2, g(x) = xe^x^2 + e^-x^2 and h(x) = x^2e^x^2 + e^-x^2. If a, b and c denote respectively, the absolute maximum of f, g and h…

    2010

  36. If the function g: (-∞, ∞) arrow (-(π)/2, (π)/2 ) is given by g(u) = 2tan^-1(e^u) - (π)/2. Then, g is

    2008

  37. The total number of local maxima and local minima of the function f(x) = begincases (2+x)^3, & -3 < x ≤ -1 \ x^2/3, & -1 < x < 2 endcases is

    2008 · with solution

  38. If f(x) = x^3 + bx^2 + cx + d and 0 < b^2 < c, then in (-∞, ∞)

    2004 · with solution

  39. The length of a longest interval in which the function 3sin x - 4sin^3 x is increasing, is

    2002 · with solution

  40. The point(s) on the curve y^3 + 3x^2 = 12y, where the tangent is vertical, is (are)

    2002 · with solution

  41. If f(x) = xe^x(1-x), then f(x) is

    2001

  42. For all x ∈ (0, 1)

    2000

  43. Let f(x) = ∫ e^x(x-1)(x-2) dx. Then, f decreases in the interval

    2000

  44. If f(x) = begincases |1-x|, & 0 < |x| ≤ 2 \ 1, & x = 0 endcases. Then, at x = 0, f has

    2000

  45. The function f(x) = sin^4 x + cos^4 x increases, if

    1999

  46. If f(x) = (x^2 - 1)/(x^2 + 1) for every real number x, then the minimum value of f

    1998 · with solution

  47. The number of values of x, where the function f(x) = cos x + cos(√(2)x) attains its maximum, is

    1998 · with solution

  48. If f(x) = x/(sin x) and g(x) = x/(tan x), where 0 < x ≤ 1, then in this interval

    1997

  49. Find the coordinates of all the points P on the ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1, for which the area of the Δ PON is maximum, where O denotes the origin and N is the foot of the perpendicular from O to the tangent…

    1996

  50. On the interval [0,1], the function x^25(1-x)^75 takes its maximum value at the point

    1995 · with solution

  51. If P(x) = a_0 + a_1x^2 + a_2x^4 + … + a_nx^2n is a polynomial in a real variable x with 0 < a_0 < a_1 < a_2 < … < a_n. Then, the function P(x) has

    1986

  52. The normal to the curve x = a(cosθ + θsinθ), y = a(sinθ - θcosθ) at any point 'θ' is such that

    1983

  53. If y = a log x + bx^2 + x has its extremum values at x = -1 and x = 2, then

    1983 · with solution

  54. If p, q and r are any real numbers, then

    1982 · with solution

  55. Let f(x) be a polynomial of degree four having extreme values at x=1 and x=2. If lim_x→ 0 [1+(f(x))/(x^2) ]=3, then f(2) is equal to

    with solution

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