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)
Your Answer
If f(x) is differentiable and strictly increasing function then the value of is
a) 1
b) 0
c) – 1
d) 2
If f(x) = xa 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
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
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
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.
b) 2
c) 3
d) 4
Find the shortest distance of the point (0, c) from the parabola y = x2, where 0 ≤ c ≤ 5.
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)
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 . . . .
Let f(x) = ∫ex (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
Let f : ℝ → ℝ be a differentiable function and f (1) = 4. Then show that the value of =
equals
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
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
