If f(x) = x^a lnx and f(0) = 0 then the value of a for which Rolle’s theorem can be applied in [0, 1] is

a) – 2

b) – 1

c) 0

d)

If f(x) is a polynomial of degree less than or equal to 2 and S be the set of all such polynomials so that

P(0) = 0

P(1) = 1, and

Then

a) S = ɸ

b) S = ax + (1 – a) x² ⩝ a ε (0, 2)

c) S = ax + (1 – a) x² ⩝ a ε (0, ∞)

d) S = ax + (1 – a) x² ⩝ a ε (0, 1)

Minimum area of the triangle formed by the tangent to the ellipse

 with co-ordinate axes is

a)

b)

c)

d) ab

Multiple choice

Let h(x) = f(x) – (f(x))² + (f(x))³ for every real number x, then

a) h increases whenever f is increasing

b) h is increasing whenever f is decreasing

c) h is decreasing whenever f is decreasing

d) nothing can be said in general

Multiple choice

The function  

has local minimum at x =

a) 0

b) 1

c) 2

d) 3

Let x and y be two real variables such that x > 0 and xy = 1. Find the minimum value of x + y.

a) 1

b) 2

c) 3

d) 4

Find the shortest distance of the point (0, c) from the parabola
y = x², where 0 ≤ c ≤ 5.

a)

b)

c)

d)

Show that   for all x ≥ 0.

Find the coordinates of the point on the curve  where the tangent to the curve has the greatest slope.

a) (0, 0)

b)

c)

d)

Find the tangents to the curve
y = cos(x + y), − 2π ≤ x ≤ 2π
that are parallel to the line x + 2y = 0

If the function f: [0, 4] → ℝ is differentiable, then for a, b ε [0, 4]

a) 8 f (a) f (b)

b) 8 f (a) f '(b)

c) 8 f '(a) f (b)

d) 8 f '(a) f '(b)

(Fill in the blanks) The function y = 2x² – ln|x| is monotonically increasing for values of x (≠0) satisfying the inequalities .  .  .  . and monotonically decreasing for values of x satisfying the inequalities .  .  .  .

a)

b)

c)

d)

Suppose f(x) is a function satisfying the following conditions

i) f(0) = 2, f(1) = 1

ii) f has a minimum value at x = 5/2 and

iii) for all x

 
where a, b are constants. Determine the constants a and b, and the function f(x).

a)

b)

c)

d)

Let f(x) = ∫e^x (x – 1) (x − 2) dx, then f(x) decreases in the interval

a) (−∞, −2)

b) (−2, −1)

c) (1, 2)

d) (2, ∞)

If f(x) be the interval of  find

a) ½

b) 1

c) 2

d) 4

Let f : ℝ → ℝ be a differentiable function and f (1) = 4. Then show that the value of   =

 equals

a)

b)

c)

d) 4 f (2)

A cubic f (x) vanishes at x = −2 and has a relative minimum/maximum at x = −1 and . If , find the cube f (x).

a) x³ + x² + x + 1

b) x³ + x² − x + 1

c) x³ − x² + x + 2

d) x³ + x² − x + 2

If  at x = π

a)

b) π

c) 2π

d) 4π

Find the limiting value of   when x

Find the limiting value of    as x a

Let f(x) be a quadratic expression which is positive for all values of x. If g(x) =  then for any real x

a) g (x) < 0

b) g (x) > 0

c) g (x) = 0

d) g (x) ≥ 0

If  then  is equal to

a)

b)

c)

d)