Application of Derivatives

The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by \({{{x^2}} \over {xy - {x^2}{y^2} - 1}}\). If the curve passes through the point (1, 1), then e . y(e) is equal to

If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the function f(x) = x2 + ax + 1 is decreasing on [1, 2], then the value of |R \(-\) S| is ____...

The number of real roots of the equation \({e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0\) is :

Let f(x) be a cubic polynomial with f(1) = \(-\)10, f(\(-\)1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = \(-\)1. Then f(3) is equal to ____________.

The number of distinct real roots of the equation 3x4 + 4x3 \(-\) 12x2 + 4 = 0 is _____________.

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the ...

A box open from top is made from a rectangular sheet of dimension a \(\times\) b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

The maximum slope of the curve \(y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x\) occurs at the point :

Let f be any function defined on R and let it satisfy the condition : \(|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R\)If f(0) = 1, then :

Let slope of the tangent line to a curve at any point P(x, y) be given by \({{x{y^2} + y} \over x}\). If the curve intersects the line x + 2y = 4 at x = \(-\)2, then the value of y, for which the point (3, y) lies on the curve, is :

Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, \(-\)3) and (4, \(-\)2\(\sqrt 2\)), and given that a \(-\) 2\(\sqrt 2\) b = 3, then (a2 + b2 + ab) is equal to _________...

Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then $...

The local maximum value of the function \(f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}\), x > 0, is

Let \(f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3\), \(x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]\). Then, f is :

If the curves, \({{{x^2}} \over a} + {{{y^2}} \over b} = 1\) and \({{{x^2}} \over c} + {{{y^2}} \over d} = 1\) intersect each other at an angle of 90\(^\circ\), then which of the following relations is TRUE?

If Rolle's theorem holds for the function \(f(x) = {x^3} - a{x^2} + bx - 4\), \(x \in [1,2]\) with \(f'\left( {{4 \over 3}} \right) = 0\), then ordered pair (a, b) is equal to :

Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = \(-\)1 and x = 1. If \(\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^3}}} = 1\), then \(5.f(2)\) is equal to _________.

If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to __________.

The function
f(x) = \({{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x\) :

The minimum value of \(\alpha\) for which the equation \({4 \over {\sin x}} + {1 \over {1 - \sin x}} = \alpha\)
has at least one
solution in \(\left( {0,{\pi \over 2}} \right)\) is .......

If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the
ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

Let \(f:R \to R\) be defined as\(f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x < - 5} \cr {2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le x \le 4} \cr {2{x^3} - 3{x^2} - 36x - 336,} & {if\,x > 4,} \cr } } \right.\)Let A = {x $$ \i...

For which of the following curves, the line \(x + \sqrt 3 y = 2\sqrt 3\) is the tangent at the point \(\left( {{{3\sqrt 3 } \over 2},{1 \over 2}} \right)\)?

If the curve y = ax2 + bx + c, x\(\in\)R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are :

Let f : R \(\to\) R be defined as\(f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr {3x{e^x},} & {x \le 0} \cr } } \right.\). Then f is increasing function in the interval

Let 'a' be a real number such that the function f(x) = ax2 + 6x \(-\) 15, x \(\in\) R is increasing in \(\left( { - \infty ,{3 \over 4}} \right)\) and decreasing in \(\left( {{3 \over 4},\infty } \right)\). Then the function g(x) = ax2 $$-$...

Let \(A = [{a_{ij}}]\) be a 3 \(\times\) 3 matrix, where \({a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , & {otherwise.} \cr } } \right.\)Let a function f...

The sum of all the local minimum values of the twice differentiable function f : R \(\to\) R defined by \(f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)\) is :

The function \(f(x) = {x^3} - 6{x^2} + ax + b\) is such that \(f(2) = f(4) = 0\). Consider two statements :Statement 1 : there exists x1, x2 \(\in\)(2, 4), x1 < x2, such that f'(x1) = \(-\)1 and f'(x2) = 0.Statement 2 : there exists x3, x4 ...

Let f : [\(-\)1, 1] \(\to\) R be defined as f(x) = ax2 + bx + c for all x\(\in\)[\(-\)1, 1], where a, b, c\(\in\)R such that f(\(-\)1) = 2, f'(\(-\)1) = 1 for x\(\in\)(\(-\)1, 1) the maximum value of f ''(x) is \({{1 \over 2}}\). If f(x) ...

Consider the function f : R \(\to\) R defined by
\(f(x) = \left\{ \matrix{ \left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfill \cr 0,\,\,x = 0 \hfill \cr} \right.\). Then f is :

Let f be a real valued function, defined on R \(-\) {\(-\)1, 1} and given by f(x) = 3 loge \(\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}\).Then in which of the following intervals, function f(x) is increasing?

The maximum value of \(f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr } } \right|,x \in R\) is...

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

Let ƒ(x) = xcos–1(–sin|x|), \(x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]\), then
which of the following is true?

Let the normal at a point P on the curve
y2 – 3x2 + y + 10 = 0 intersect the y-axis at \(\left( {0,{3 \over 2}} \right)\)
. If m is the slope of the tangent at P to
the curve, then |m| is equal to

If c is a point at which Rolle's theorem holds
for the function,
f(x) = \({\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)\) in the
interval [3, 4], where a \(\in\) R, then ƒ''(c) is equal
to

The length of the perpendicular from the origin,
on the normal to the curve, x2 + 2xy – 3y2 = 0
at the point (2,2) is

Let ƒ(x) be a polynomial of degree 3 such that
ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point
at x = –1 and ƒ'(x) has a critical point at x = 1.
Then ƒ(x) has a local minima at x = _______.

Let the function, ƒ:[-7, 0]\(\to\)R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = -
3 and ƒ'(x) \(\le\) 2, for all x \(\in\) (-7,0), then for all such functions ƒ, ƒ(-1) + ƒ(0) lies in the interval:

Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points.
If \(\mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} \over {{x^3}}}} \right) = 4\), then which one of the following is not true?

The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3
- 4x2
+ 8x + 11,
when x \(\in\) [0, 1] is:

The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real
numbers greater than 1. Then the average speed of the car over the time interval [t1
, t2
] is
attained at the point :

The set of all real values of \(\lambda\) for which the
function
\(f(x) = \left( {1 - {{\cos }^2}x} \right)\left( {\lambda + \sin x} \right),x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)\)has exactly one maxima and exactly one
m...

If the tangent to the curve, y = f (x) = xloge x,
(x > 0) at a point (c, f(c)) is parallel to the
line-segment joining the points (1, 0) and
(e, e), then c is equal to :

If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, -4), then PQ2 is equal to:

Which of the following points lies on the
tangent to the curve
x4ey + 2\(\sqrt {y + 1}\) = 3 at the
point (1, 0)?

If x = 1 is a critical point of the function
f(x) = (3x2
+ ax – 2 – a)ex
, then :

If the lines x + y = a and x – y = b touch the
curve y = x2
– 3x + 2 at the points where the
curve intersects the x-axis, then \({a \over b}\) is equal
to _______.