If f ( x ) = a log e | x | + b x 2 + x has extremums at x = 1 and x = 3 , then
If a, b, c are real numbers, then the intervals in which f ( x ) = x + a 2 a b a c a b x + b 2 b c a c b c x + c 2 is strictly decreasing
Use differential to approximate ( 27.3 ) 1 / 3
The function g ( x ) = e x 2 log ( π + x ) log ( e + x ) ( x ≥ 0 ) is
The function f satisfying f ( b ) − f ( a ) b − a ≠ f ′ ( x ) for any x ∈ ( a , b ) is
The equation e x − 1 + x − 2 = 0 as
If ϕ ( x ) is a polynomial function and ϕ ′ ( x ) > ϕ ( x ) ∀ x ≥ 1 and ϕ ( 1 ) = 0 , then
Let, then f ( x ) = xcos – 1 ( – sin | x | ) , x ∈ – π 2 , π 2 which of the following is true?
Let the normal at a point P on the curve y 2 – 3 x 2 + y + 10 = 0 intersect the y -axis at 0 , 3 2 . If m is the slope of the tangent at P to the curve, then | m | is equal to
The curve x + y − log e ( x + y ) = 2 x + 5 has a vertical tangent at the point ( α , β ) . Then α + β =
Find the approximate change in the volume V of a cube of side x meters caused by increasing side by 1%.
The constant θ of Lagrange’s theorem for f ( x ) = x 2 − 2 x + 3 is 1 , 3 2 is
The coordinates of the point P ( x , y ) lying in the first quadrant on the ellipse x 2 / 8 + y 2 / 18 = 1 so that the area of the triangle formed by the tangent at P and the coordinate axes is the smallest, are given by
A quadratic function y=f(x) if it touches the line y=x at the point x=1 and passes through the point (-1, 0)
Use differential to approximate ( 27.3 ) 1 / 3
Let f ( x ) = x e a x ; x ≤ 0 x + a x 2 − x 3 ; x > 0 where a ‘ is a positive constant. Find the interval in which f ′ ( x ) is increasing.
If f ( x ) = x 2 + 2 b x + 2 c 2 a n d g ( x ) = − x 2 − 2 c x + b 2 are such that min f(x) > max g(x), then the relation between b and c is
If f ( x ) = x + sin x , g ( x ) = e − x , u = c + 1 − c , v = c − c − 1 , ( c > 1 ) , then
The function f x = sin 4 x + cos 4 x increases if
L e t A = sin x + tan x a n d B = 2 x i n t h e i n t e r v a l 0 < x < π 2 then
The number of points on the curve y = x 3 – 3 x at which tangent is parallel to x-axis
The maximum value of f x = sin 2 x 1 + cos 2 x cos 2 x 1 + sin 2 x cos 2 x cos 2 x sin 2 x cos 2 x sin 2 x , x ∈ R is:
If the tangent at any point P 4 m 2 , 8 m 3 of x 3 − y 2 = 0 is also a normal to curve x 3 − y 2 = 0 , then m =
If the slope of line through the origin is tangent to the curve y = x 3 + x + 16 is m ,then the value of m is
If the curves x 2 a 2 + y 2 4 = 1 and y 2 = 16 x intersect at right angle, then
If the function f ( x ) = x 4 + b x 2 + 8 x + 1 has a horizontal tangent and a point x for f ′′ ( x ) = 0 then the value of b is equal to
If water is poured into an inverted hollow cone whose semi-vertical angle is 30° , then the its depth (measured along axis) increases at the rate of 1cm/sec, the rate of which the volume of water increases when the depth is 24 cm.
Use differential to approximate 16.6
If the approximate value of ( 1.999 ) 6 is
If a, b, c are real numbers, then the intervals in which f x = x + a 2 a b a c a b x + b 2 b c a c b c x + c 2 is strictly decreasing
The slope of the tangent to the curve represented by x = t 2 + 3 t − 8 and y = 2 t 2 − 2 t − 5 at the point M ( L , − 1 ) is
The curve y = a x 3 + b x 2 + c x + 8 touches x -axis at P ( − 2 , 0 ) and cuts the y-axis at a point Q where its gradient is 3. The values of a, b, c are respectively
On the curve y = x 3 , the interval of values of x for which abscissa changes at a faster rate than the ordinate.
Let x be the length of one of the equal sides of an isosceles triangle and let θ be the angle between them. If x is increasing at the rate of ( 1 / 12 ) m / hr and θ is increasing at the rate of π 180 radians / hr , then the rate in m 2 / hr at which the area of the triangle is increasing when x = 12 m and θ = π 4
A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of when the thickness of ice is 5 cm than the rate at which the thickness of ice decreases.
Show that 1 + x log e x + x 2 + 1 ≥ 1 + x 2 for all x ≥ 0
Use the function f ( x ) = x 1 / x , x > 0 , to determine the bigger of the two numbers e π and π e .
Let the x co-ordinate of a point P on the function f ( x ) = 2 x ( x − 3 ) n o n [ 0 , 3 ] for which Rolle’s theorem is valid is 3 4 . Then find 100n is,
The tangent to the curve x = a cos 2 θ cos θ , y = a cos 2 θ sin θ at the point corresponding to θ = π / 6 is
Tangent is drawn to ellipse x 2 27 + y 2 = 1 a t ( 3 3 cos θ , sin θ ) ( where θ ∈ ( 0 , π / 2 ) ) . Then the value of such that sum of intercepts on axes made by this tangent is least is
Let f ( x ) = ( x − 3 ) 5 ( x + 1 ) 4 then
The number of solutions of the equation x 3 + 2 x 2 + 5 x + 2 cos x = 0 in [ 0 , 2 π ] is
Let the function f : − 7 , 0 R be continuous on − 7 , 0 and differentiable on − 7 , 0 . If f − 7 = − 3 and f ‘ x ≤ 2 for all such function f, f − 1 + f 0 is in the interval:
Let f x be a polynomial of degree 5 such that x = ± 1 are its critical points. If lim x 0 2 + f x x 3 = 4 , then which one of the following is not true?
The value of c in the Lagrange’s mean value theorem for the function f x = x 3 − 4 x 2 + 8 x + 11 , when x ∈ 0 , 1 is:
If c is a point at which Rolle’s theorem holds for the function, f ( x ) = log e x 2 + α 7 x in the interval [ 3 , 4 ] , where α ∈ R , then f ‘ ‘ ( c ) is equal to:
Let f ( x ) be any function continuous and twice differentiable function on ( a , b ) . If for all x ∈ ( a , b ) , f ‘ ( x ) > 0 and f ‘ ‘ ( x ) < 0 , then for any c ∈ ( a , b ) , f ( c ) – f ( a ) f ( b ) – f ( c ) is greater than
A spherical iron ball of 10cm radius is located with a layer of ice of uniform thickness that melts at a rate of 50 c m 3 / min . When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of ice decreases, is:
The height of the cylinder of maximum volume that can be inscribed in a sphere of radius ‘a’ is
If the curves y = x 3 + a x and y = b x 2 + c pass through the point ( – 1 , 0 ) and have a common tangent line at this point, then the value of a – 3 b + c =
The equation of the common tangent to the circle ( x – 3 ) 2 + y 2 = 9 and the parabola y 2 = 4 x , above the x-axis is
If y = 2 x – tan – 1 x – log x + 1 + x 2 , then:
The function f ( x ) = tan – 1 ( sinx + cosx ) is an increasing function in
If f ( x ) = 2 x + cot – 1 x + log 1 + x 2 – x , then f ( x )
Sixteen meter of wire is available to fence off a flower bed in the form of a sector of a circle. If the flower bed has the maximum surface area, then the radius is
The height of the right circular cylinder of maximum volume that can be inscribed in a sphere of radius ‘a’ is
Rolle’s theorem is applicable to function f ( x ) = ln x x over [ a , b ] where a , b are positive integers, then the value of a + b
The length of the sub-tangent to the hyperbola x 2 – 4 y 2 = 4 c o r r e s p o n d i n g t o t h e n o r m a l h a v i n g s l o p e u n i t y i s 1 k , t h e n k i s e q u a l t o
A solid rectangular brick is to be made from 1 cubic feet of clay. The length of the brick must be 3 times as long as its width. The width of brick for which it will have minimum surface area is then a , is a 3
The least positive integral value of ‘a’ such that 2x+ a x 2 ≥ 6 , ∀ x ∈ R i s
Let f ( x ) satisfy all the conditions of Lagrange’s mean value theorem in [ 0 , 2 ] . If f ( 0 ) = 0 and f ′ ( x ) ≤ 1 2 for all x in [ 0 , 2 ] , then
If the slope of line through the origin is tangent to the curve y = x 3 + x + 16 is m , then the value of m is
If a < ( 28 ) 1 3 − 3 < b , then ( a , b ) is
The equations of the tangents at the origin to the curve y 2 = x 2 1 + x + x 2 is
The curve y = x 3 + x 2 − x has two horizontal tangents. The distance between these two horizontal lines is
Let f : R R , f ( 0 ) = 0 , f ′ ( x ) > 0 and f ′′ ( x ) > 0 and A α , f α , B β , f β , C γ , f γ be three points on the graph of y = f ( x ) and 0 < α < β < γ . Then which of the following is true?
If M x o , y o is the point on the curve 3 x 2 − 4 y 2 = 72 which is nearest to the line 3 x + 2 y + 1 = 0 , then the value of x o + y o is equal to
If f ( x ) = 2 | x | + | x + 2 | − | | x + 2 | − 2 | x ∥ then no. of points of local minima / maxima is
If g ( x ) = 2 f 2 x 3 − 3 x 2 + f 6 x 2 − 4 x 3 − 3 , ∀ x ∈ R and f ′′ ( x ) > 0 , ∀ x ∈ R , then g ( x ) is incresing in
The point on the graph of the curve y = ( x − 3 ) 2 , where the tangent is parallel to the line joining of (3,0), (4, 1) is
Let a , b , c , d are non − zero real numbers such that 6 a + 4 b + 3 c + 3 d = 0 , then the equation a x 3 + b x 2 + c x + d = 0 has
Maximum slope of the curve y = − x 3 + 3 x 2 + 9 x − 27 is
If a particle moves along a straight line by S = 4 t 2 − 8 t + 3 and the time at which the particle comes to rest is ‘ t ‘ seconds ‘ 2 t ‘ is
The length of the longest interval is which the function 3 sin x − 4 sin 3 x is increasing is π k . Then k 2 =
If f ( x ) = sin − 1 x 2 + cos − 1 x 2 is stationary at x = a . Then a 2 =
A point is moving along the cubical parabola 12 y = x 3 . The rate of ordinate is less than the rate of abscissa when
The approximate value of f ( 5.001 ) , where f ( x ) = x 3 − 7 x 2 + 15 .
A point on the curve y = ( x − 3 ) 2 , where the tangent is parallel to the chord joining the points ( 3 , 0 ) and ( 4 , 1 )
The real number k for which the equation, 2 x 3 + 3 x + k = 0 has two distinct real roots in [ 0 , 1 ]
If 0 < a < b < π b and f ( a , b ) = tan b − tan a b − a , then
A man is walking of the rate of 5 km/h towards the foot of a tower 16 m high. The roots at which he is approaching the top when he is 12 m from the tower is
An aeroplane is flying horizontally at a height of 2 3 km with a velocity of 15 km/hr. The rate at which it is receding from a fixed point on the ground which is passed over 2 minutes ago.
The angle between the parabolas y 2 = 4 a x and x 2 = 4 a y at their point of intersection other than the origin.
A quadratic function y=f(x) if it touches the line y=x at the point x=1 and passes through the point (-1, 0)
Use the function f ( x ) = x 1 / x , x > 0 , to determine the bigger of the two numbers e π and π e .
Find the approximate change in the volume V of a cube of side ‘x’ meters caused by increasing the side by 3%
Use differential to approximate ( 27.3 ) 1 / 3
A dynamic blast blows a heavy rock straight up with a launch velocity of 160 m/sec. It reaches a height of after t sec. The velocity of the rock when it is 256 m above the ground on the way up is
If the radius of a sphere is measured as 10cm with an error of 0.01 cm, then find the approximate error in calculating its volume.
The coordinates of the point P on the curve y 2 = 2 x 3 , the tangent at which is perpendicular to the line 4x – 3y+2 = 0, are given by
The point(s) on the curve y 3 + 3 x 2 = 12 y where the tangent is vertical is (are)
The equation of the common tangent to the curves y 2 = 8 x and x y = − 1 is
If the tangent at (1, 1) on y 2 = x ( 2 − x ) 2 meets the curve again at P, then P is
If water is poured into an inverted hollow cone whose semi-vertical angle is 30° , then the its depth (measured along axis) increases at the rate of 1cm/sec, the rate of which the volume of water increases when the depth is 24 cm.
An aeroplane is flying horizontally at a height of km with a velocity of 15 km/hr. The rate at which it is receding from a fixed point on the ground which is passed over 2 minutes ago.
f (x) is continuous on [0, 2] , differentiable on (0,2) , f(0) = 2 , f(2) = 8 and f ‘ ( x ) ≤ 3 for all x in (0,2) , then the value of f(1)
The angle between the parabolas y 2 = 4 a x a n d x 2 = 4 a y at their point of intersection other than the origin.
The points of contact of the vertical tangents to x = 2 − 3 sin θ , y = 3 + 2 cos θ a r e
If f ( x ) = x sin x a n d g ( x ) = x tan x w h e r e 0 < x ≤ 1 , then in this interval
The length of a longest interval in which the function 3 sin x − 4 sin 3 x is incresing is
Let f ( x ) = x e x ( 1 − x ) , then f ( x ) is
Suppose f is differentiable on R and a ≤ f ′ ( x ) ≤ b for all x ∈ R where a , b > 0 . If f ( 0 ) = 0 , then
The minimum value of f ( x ) = | 3 − x | + | 2 + x | + | 5 − x | is
If f ( x ) = x 2 − 1 x 2 + 1 , for every real number, then minimum value of f
The coordinates of the point on the parabola y 2 = 8 x which is at minimum distance from the circle x 2 + ( y + 6 ) 2 = 1 are
The image of the interval [ − 1 , 3 ] under the maping f ( x ) = 4 x 3 − 12 x is
The difference between the greatest and least values of the function f ( x ) = cos x + 1 2 cos 2 x − 1 3 cos 3 x i s
If y = a log | x | + b x 2 + x has its extremum values at x = − 1 and x = 2 , then
If θ is the angle (semi-vertical) of a cone of maximum volume and given slant height, then tan θ is given by
Let f ( x ) = ( 1 + x ) n − ( 1 + n x ) , x ∈ [ − 1 , ∞ ) , then f
The tangent to the curve y = e x drawn at the point c , e c intersects the line joining the points c − 1 , e c − 1 and c + 1 , e c + 1
The function y = x 1 + x 2 decreases in the interval
f ( x ) = ( x − 8 ) 4 ( x − 9 ) 5 , 0 ≤ x ≤ 10 , monotonically decreases in
If f ( x ) = xe x ( x − 1 ) , then f(x) is
If f ( x ) = kx 3 − 9 x 2 + 9 x + 3 is monotonically increasing in R, then
If the function f ( x ) = Ksin x + 2 cos x sin x + cos x is strictly increasing for all values of x, then
Let f : R R be a function such that f ( x ) = ax + 3 sin x + 4 cos x .Then f(x) is invertible if
Let f : R R be a differentiable function for all values of x and has the property that f (x) and f ‘(x) have opposite signs for all values of x. Then,
Let f : R R be a differentiable and increasing function ∀ x ∈ R . If the tangent drawn to the curve at any point x ∈ ( a , b ) always lies below the curve, then
Let f (x) be a function such that f ′ ( x ) = log 1 / 3 log 3 ( sin x + a ) .If f(x) is decreasing for all real values of x, then
lf f ( x ) = x 3 + 4 x 2 + λx + 1 is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then
f ( x ) = xlog e x monotonically decreases in
The set of value(s) of a for which the function f ( x ) = ax 3 3 + ( a + 2 ) x 2 + ( a − 1 ) x + 2 possesses a negative point of inflection is
The maximum value of the function of f ( x ) = ( 1 + x ) 0 .6 1 + x 0 .6 in the interval [0, 1] is
A function s(x) is defined as g ( x ) = 1 4 f 2 x 2 − 1 + 1 2 f 1 − x 2 and f ′ ( x ) is an increasing function. Then g(x) is increasing in the interval
If f ′′ ( x ) > 0 ∀ x ∈ R , f ′ ( 3 ) = 0 , and g ( x ) = f tan 2 x − 2 tan x + 4 , 0 < x < π 2 , then g(x) is increasing in
If A > 0 , B > 0 , and A + B = π 3 , then the maximum value of tan A tan B is
The function sin ( x + a ) sin ( x + b ) ,has no maxima or minima if
The tangent line at (2, 4) to the curve y = x 3 − 3 x + 2 meets the x-axis at
A point on the curve y = x 3 – 3 x + 5 at which the tangent line is parallel to y = – 2 x is
If y = x – log ( 1 + x ) , then minimum value of y is
The point of inflection of y = x 3 – 5 x 2 + 3 x – 5 is
The minimum rate of change of the function f ( x ) = 3 x 5 – 5 x 3 + 5 x – 7 is
