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Application of Derivatives

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  1. 101.If x=-1 and x=2 are extreme points of [f( x ) = , | x |+ x^2 + x] then
  2. 102.Let f(x) be a polynomial of degree four having extreme values at x=1 and x=2.
  3. 103.The normal to the curve, x^2 + 2xy - 3y^2 = 0, at (1,1)
  4. 104.Consider : f [( x ) = ^ - 1( 1 + x 1 - x ),x ( 0, 2 ).] A normal to y = f[( x )] at [x = 6…
  5. 105.A wire of length 2 units is cut into two parts which are bent respectively to form a squar…
  6. 106.Let C be a curve given by y(x) = 1 + [ 4x - 3 ,x > 3 4.] If P is a point on C, such that t…
  7. 107.Let f(x) = sin4x + cos4 x.
  8. 108.If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3-1, t [ ] …
  9. 109.The normal to the curve y(x -- 2)(x -- 3) = x + 6 at the point where the curve intersects …
  10. 110.Twenty meters of wire is available for fencing off a flower-bed in the form of a circular …
  11. 111.The tangent at the point (2, -2) to the curve, x2y2 - 2x = 4(1 - y) does not pass through …
  12. 112.A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B.
  13. 113.The function f defined by f(x) = x3 - 3x2 + 5x + 7 , is :
  14. 114.If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value…
  15. 115.Let [f( x ) = x^2 + 1 x^2] and [g( x ) = x - 1 x], [x R - - 1,0,1 ].
  16. 116.If [ ] is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points…
  17. 117.Let M and m be respectively the absolute maximum and the absolute minimum values of the fu…
  18. 118.If the tangent to the curve [y = x x^2 - 3] , [x ,( x 3 )], at a point ([ ], [ ]) [ ] (0, …
  19. 119.A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness t…
  20. 120.The tangent to the curve, y = xex2 passing through the point (1, e) also passes through th…
  21. 121.A helicopter is flying along the curve given by y -- x3/2 = 7, (x [ ] 0).
  22. 122.The shortest distance between the point [( 3 2,0 )] and the curve y = [ x ], (x > 0), is -
  23. 123.Let f(x) = [x a^2 + x^2 - d - x b^2 + ( d - x )^2 ,,,] x [, ] R, where a, b and d are non-…
  24. 124.The maximum value of the function f(x) = 3x3 -- 18x2 + 27x -- 40 on the set S = [ x, R:x^2…
  25. 125.A 2 m ladder leans against a vertical wall.
  26. 126.If m is the minimum value of k for which the function f(x) = x[ kx - x^2 ] is increasing i…
  27. 127.If the function f given by f(x) = x3 -- 3(a -- 2)x2 + 3ax + 7, for some a[ ]R is increasin…
  28. 128.The tangent to the curve y = x2 -- 5x + 5, parallel to the line 2y = 4x + 1, also passes t…
  29. 129.Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is [2y x^2].
  30. 130.Let f : [0, 2] [ ] R be a twice differentiable function such that f''(x) > 0, for all x [ …
  31. 131.If S1 and S2 are respectively the sets of local minimum and local maximum points of the fu…
  32. 132.A water tank has the shape of an inverted right circular cone, whose semi-vertical angle i…
  33. 133.If the tangent to the curve, y = x3 + ax -- b at the point (1, --5) is perpendicular to th…
  34. 134.Let S be the set of all values of x for which the tangent to the curve y = f(x) = x3 -- x2…
  35. 135.The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :
  36. 136.The equation of the normal to the curve y = (1+x)2y + cos 2(sin--1x) at x = 0 is :
  37. 137.Let f : (--1, [ ]) [ ] R be defined by f(0) = 1 and f(x) = [1 x _e( 1 + x )], x [ ] 0.
  38. 138.If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a lo…
  39. 139.If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joinin…
  40. 140.Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x -- 3.
  41. 141.If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape;…
  42. 142.The function, f(x) = (3x -- 7)x2/3, x [ ] R, is increasing for all x lying in :
  43. 143.The area (in sq.
  44. 144.Which of the following points lies on the tangent to the curve x4ey + 2[ y + 1 ] = 3 at th…
  45. 145.If x = 1 is a critical point of the function f(x) = (3x2 + ax -- 2 -- a)ex , then :
  46. 146.If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, -4), then PQ2 …
  47. 147.The set of all real values of [ ] for which the function [f(x) = ( 1 - ^2x )( + x ),x ( - …
  48. 148.If the tangent to the curve, y = f (x) = xloge x, (x > 0) at a point (c, f(c)) is parallel…
  49. 149.The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b …
  50. 150.Let f(x) be a polynomial of degree 5 such that x = 1 are its critical points.