If f ( x ) = a log e | x | + b x 2 + x has extremums at x = 1 and x = 3 , then

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?

The constant θ of Lagrange’s theorem for f ( x ) = x 2 − 2 x + 3 is 1 , 3 2 is

Use differential to approximate ( 27.3 ) 1 / 3

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

If the curves x 2 a 2 + y 2 4 = 1 and y 2 = 16 x intersect at right angle, then

Use differential to approximate 16.6

If the approximate value of ( 1.999 ) 6 is

Show that 1 + x log e x + x 2 + 1 ≥ 1 + x 2 for all x ≥ 0

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

The height of the cylinder of maximum volume that can be inscribed in a sphere of radius ‘a’ 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 )

The least positive integral value of ‘a’ such that 2x+ a x 2 ≥ 6 , ∀ x ∈ R i s

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

Maximum slope of the curve y = − x 3 + 3 x 2 + 9 x − 27 is

If f ( x ) = sin − 1 x 2 + cos − 1 x 2 is stationary at x = a . Then a 2 =

The approximate value of f ( 5.001 ) , where f ( x ) = x 3 − 7 x 2 + 15 .

If 0 < a < b < π b and f ( a , b ) = tan b − tan a b − a , then

Use differential to approximate ( 27.3 ) 1 / 3

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

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

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 image of the interval [ − 1 , 3 ] under the maping f ( x ) = 4 x 3 − 12 x is

If y = a log | x | + b x 2 + x has its extremum values at x = − 1 and x = 2 , then

Let f ( x ) = ( 1 + x ) n − ( 1 + n x ) , x ∈ [ − 1 , ∞ ) , then f

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

Let f : R R be a function such that f ( x ) = ax + 3 sin x + 4 cos x .Then f(x) is invertible if

f ( x ) = xlog e x monotonically decreases in

The maximum value of the function of f ( x ) = ( 1 + x ) 0 .6 1 + x 0 .6 in the interval [0, 1] is

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