# Continuity and Differentiability

This is an MCQ-based quiz on Continuity and Differentiability.

This includes Continuity, Differentiability, Exponential, Logarithmic Functions, Logarithmic Differentiation, Derivatives of Functions in Parametric Forms, Second Order Derivative, and Mean Value Theorem.

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Let f(x) = |sin x| Then

f is everywhere differentiable f is everywhere continuous but not differentiable at x = nπ, n ∈ Z f is everywhere continuous but no differentiable at x = (2n + 1) *π / 2 n ∈ Z None of these

The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval [o, √3] is

1 -1 3/2 1/3

The function f(x) = e|x| is

Continuous everywhere but not differentiable at x = 0

Continuous and differentiable everywhere

Not continuous at x = 0

None of these

For x ∈ R, f(x) = |log 2 – sin x| and g(x) =f(f(x)), then

g is not differentiable at x = 0 g'(0) = cos (log 2) g'(0) = -cos (log 2) g is differentiable at x = 0 and g'(0) = – sin (log 2).

If g is the inverse of a function f and f"(x) = 1 /(1+x^5), then g"(x) is equal to

5x^4 1/(1+g(x)^5) 1 + {g(x)}^5 1 + x^5

If y = sec (tan^-1 x), then dy /dx at x = 1 is equal to

1/2 1 √2 1/√2

Let f : (- 1, 1) → R be a differentiable function with f(0) = – 1 and f"(0) = 1. Let g(x) = [f (2f(x) + 2)]². Then g"(0) =

4 -4 log 2 -log 2.

If f: R → R is a function defined byf(x) = [x] cos ((2x-1) / 2)π, where [x] denotes the greatest integer function, then ‘f’ is

Continuous for every real x

Discontinuous only at x = 0

Discontinuous only at non-zero integral values of x

Continuous only at x = 0.

If sin y + e^-x *cos y = e, then dy/dx at (1, π) is equal to

sin y -x cos y e sin y – x cos y

The derivative of y = (1 – x) (2 – x)…. (n – x) at x = 1 is equal to

0 (-1) (n – 1)! n ! – 1 (-1)n^-1 (n – 1)!

If y = (1 + x) (1 + x²) (1 + x^4) …….. (1 + x^2n), then the value of dy/dx at x = 0 is

0 -1 1 None of these

If f(x) = 5x / (1-x)^(2/3) + cos² (2x + 1), then f"(0) =

5 + 2 sin 2 5 + 2 cos 2 5 – 2 sin 2 5 – 2 cos 2

If y = ax² + b, then dy/dx at x = 2 is equal to ax

4a 3a 2a None of these

The set of points where the function f given by f (x) =| 2x – 1| sin x is differentiable is

R

R-1 / 2

(0, ∞)

None of these

The function f(x) = (4−x^2) / (4x−x^3) is

Discontinuous at only one point at x = 0

Discontinuous at exactly two points

Discontinuous at exactly three points

None of these

Quiz/Test Summary
Title: Continuity and Differentiability
Questions: 15
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