Application of Derivatives

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) \(\ge\) 1 and f''(x) \(\ge\) 4, for all x \(\in\) (1, 6), then :

The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x2–1 below the x-axis, is :

The function, f(x) = (3x – 7)x2/3, x \(\in\) R, is
increasing for all x lying in :

If the surface area of a cube is increasing at a
rate of 3.6 cm2/sec, retaining its shape; then
the rate of change of its volume (in cm3/sec),
when the length of a side of the cube is
10 cm, is :

Let P(h, k) be a point on the curve
y = x2
+ 7x + 2, nearest to the line, y = 3x – 3.
Then the equation of the normal to the curve at
P is :

If the tangent to the curve y = x + sin y at a point
(a, b) is parallel to the line joining \(\left( {0,{3 \over 2}} \right)\) and \(\left( {{1 \over 2},2} \right)\), then :

If p(x) be a polynomial of degree three that has
a local maximum value 8 at x = 1 and a local
minimum value 4 at x = 2; then p(0) is equal to :

The equation of the normal to the curve
y = (1+x)2y + cos
2(sin–1x) at x = 0 is :

Let f : (–1,
\(\infty\))
\(\to\) R be defined by f(0) = 1 and
f(x) = \({1 \over x}{\log _e}\left( {1 + x} \right)\), x \(\ne\) 0. Then the function f :

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

If ƒ(x) is a non-zero polynomial of degree four,
having local extreme points at x = –1, 0, 1; then
the set
S = {x \(\in\) R : ƒ(x) = ƒ(0)}
Contains exactly :

Let S be the set of all values of x for which the
tangent to the curve
y = ƒ(x) = x3 – x2 – 2x at
(x, y) is parallel to the line segment joining the
points (1, ƒ(1)) and (–1, ƒ(–1)), then S is equal
to :

If the tangent to the curve, y = x3 + 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 ?

A water tank has the shape of an inverted right
circular cone, whose semi-vertical angle is
\({\tan ^{ - 1}}\left( {{1 \over 2}} \right)\). Water is poured into it at a constant
rate of 5 cubic meter per minute. The the rate
(in m/min.), at...

Let ƒ : [0, 2] \(\to\) R be a twice differentiable
function such that ƒ''(x) > 0, for all x \(\in\) (0, 2).
If \(\phi\)(x) = ƒ(x) + ƒ(2 – x), then \(\phi\) is :

If S1 and S2 are respectively the sets of local
minimum and local maximum points of the function,
ƒ(x) = 9x4 + 12x3 – 36x2 + 25, x \(\in\) R,
then :

Given that the slope of the tangent to a curve y
= y(x) at any point (x, y) is
\(2y \over x^2\). If the curve passes through the centre of the circle x2 + y2 – 2x – 2y = 0, then its equation is :

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

If the function f given by f(x) = x3 – 3(a – 2)x2 + 3ax + 7, for some a\(\in\)R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, $${{f\left( x \right) - 14} \over {{{\left( {x - 1} \right)}^2}}} = 0\left( {...

The tangent to the curve y = x2 – 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point :

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

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate
25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the
horizontal ground when t...

The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = \(\left\{ {x\, \in R:{x^2} + 30 \le 11x} \right\}\) is :

Let f(x) = \({x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,\) x \(\, \in\) R, where a, b and d are non-zero real constants. Then :

The shortest distance between the point  \(\left( {{3 \over 2},0} \right)\)   and the curve y = \(\sqrt x\), (x > 0), is -

A helicopter is flying along the curve given by y – x3/2 = 7, (x \(\ge\) 0). A soldier positioned at the point \(\left( {{1 \over 2},7} \right)\) wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is ...

The tangent to the curve, y = xex2 passing through the point (1, e) also passes through the point

If the tangent to the curve \(y = {x \over {{x^2} - 3}}\)
, \(x \in \rho ,\left( {x \ne \pm \sqrt 3 } \right)\), at a point (\(\alpha\), \(\beta\)) \(\ne\) (0, 0) on it is parallel to the line
2x + 6y – 11 = 0, then :

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 ice
decreases, is :...

Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x3 \(-\) 9x2 + 12x + 5 in the interval [0, 3]. Then M \(-\)m is equal to :

If \(\beta\) is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points (3 cos \(\theta\), \(\sqrt 3 \sin \theta\)) and (\(-\) 3 sin \(\theta\), \(\sqrt 3 \,\cos \theta\)); $$\theta \in \left( {0,{\pi \over ...

If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm2) of this cone is :

Let \(f\left( x \right) = {x^2} + {1 \over {{x^2}}}\) and \(g\left( x \right) = x - {1 \over x}\),
\(x \in R - \left\{ { - 1,0,1} \right\}\).
If \(h\left( x \right) = {{f\left( x \right)} \over {g\left( x \right)}}\), then the local minimu...

If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is :

The function f defined by
f(x) = x3 \(-\) 3x2 + 5x + 7 , is :

A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point :

The tangent at the point (2, \(-\)2) to the curve, x2y2 \(-\) 2x = 4(1 \(-\) y) does not pass through the point :

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the
maximum area (in sq. m) of the flower-bed, is :

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 :

If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3−1, t \(\in\) R, meets the curve again at a point Q, then the coordinates of Q are :

The minimum distance of a point on the curve y = x2−4 from the origin is :

Let C be a curve given by y(x) = 1 + \(\sqrt {4x - 3} ,x > {3 \over 4}.\) If P is a point
on C, such that the tangent at P has slope \({2 \over 3}\), then a point through which the normal at P passes, is :

Let f(x) = sin4x + cos4 x. Then f is an increasing function in the interval :

Consider :
f \(\left( x \right) = {\tan ^{ - 1}}\left( {\sqrt {{{1 + \sin x} \over {1 - \sin x}}} } \right),x \in \left( {0,{\pi \over 2}} \right).\)
A normal to \(y =\) f\(\left( x \right)\) at \(x = {\pi \over 6}\) also passes through ...

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, then:

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

Let \(f(x)\) be a polynomial of degree four having extreme values
at \(x=1\) and \(x=2\). If \(\mathop {\lim }\limits_{x \to 0} \left[ {1 + {{f\left( x \right)} \over {{x^2}}}} \right] = 3\), then f\((2)\) is equal to :

If \(x=-1\) and \(x=2\) are extreme points of \(f\left( x \right) = \alpha \,\log \left| x \right|+\beta {x^2} + x\) then

If \(f\) and \(g\) are differentiable functions in \(\left[ {0,1} \right]\) satisfying
\(f\left( 0 \right) = 2 = g\left( 1 \right),g\left( 0 \right) = 0\) and \(f\left( 1 \right) = 6,\) then for some \(c \in \left] {0,1} \right[\)

The intercepts on \(x\)-axis made by tangents to the curve,
\(y = \int\limits_0^x {\left| t \right|dt,x \in R,}\) which are parallel to the line \(y=2x\), are equal to :