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)
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
a)
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 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 . . . .
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).
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