A function f : ℝ → ℝ satisfies the equation
f(x + y) = f(x) . f(y) x, y in ℝ and f(x) ≠ 0 for any x in ℝ. Let the function be differentiable at x = 0 and . Show that. Hence determine f(x).
a) ex
b) e2x
c) 2ex
d) 2e2x
Your Answer
Let f (x) be a continuous function satisfying If exists, find its value.
a) 0
b) 1
c) 2
d) 4
Let ℝ be the set of real numbers and f : ℝ → ℝ such that for all x and y in ℝ, . Then f (x) is a constant.
a) True
b) False
Find
b) e
c) ez
d) e3
Let
Determine a and b so that f is continuous at x = 0.
a)
b)
c)
d)
Test whether
f(x) is continuous at x = 0
f(x) is differentiable at x = 0
a) f(x) is differentiable and continuous at x = 0
b) f(x) is continuous but not differentiable at x = 0
c) f(x) is neither continuous nor differentiable at x = 0
If a function f : is an odd function such that for x ε [a, 2a] and the left hand derivative at
x = a is 0 then find the left hand derivative at x =
c) a
d) 2a
P(x) is a polynomial function such that P(1) = 0, > P(x)
x > 1. Then x > 1,
a) P(x) > 0
b) P(x) = 0
c) P(x) < 1
If and = and f(0) = 0. Find the value of . Given that 0 < <
d) 1
f(x) is a function such that and the tangent at any point passes through (1, 2). Find the equation of the tangent.
a) x = 2
b) y = 2
c) x + y = 2
d) x – y = 2
If exists then both the limits and exist
A = is equal to
Let f(x) =
If f is continuous for all x, then k is equal to
a) 3
b) 5
c) 7
d) 9
Identify a discontinuous function y = f(x) satisfying
For the function
The derivative from right . . . . and the derivative from the left . . . .
a) 0, 0
b) 0, 1
c) 1, 0
d) 1, 1
Then
L = = . . . .
a) – 1
b) 0
c) 1
d) 2
If f(9) = 9, then equals
c) e3
d) e5
Let f(x) = x|x|. The set of points where f(x) is twice differentiable is . . . .
a) ℝ
c) ℝ − {0, 1}
c) e
d) e2
Match the following
Let the function defined in column 1 has domain
Column 1
Column 2
i) x + sinx
A)increasing
ii) secx
B) decreasing
C)neither increasing nor decreasing
a) i) → A, ii) → B
b) i) → A, ii) → C
c) i) → C, ii) → A
d) i) → B, ii) → C
If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) = for all real x where k is a real constant.
The line y = x meets y = for k ≤ 0 at
a) No point
b) One point
c) Two points
d) More than two points
