Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 29

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A function $f : R \to R$ is defined as

$f (x) = \lim_{n \to \infty} \frac{a x^{2} + b x + c + e^{n x}}{1 + c \cdot e^{n x}}$ , where $f$ is continuous on $R$ , then

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Important Questions on Continuity and Differentiability

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Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 30

For every pair of continuous functions

f, g : [0, 1] \to R

such that

\max \{f (x) : x \in [0, 1]\} = \max \{g (x) : x \in [0, 1]\}

the correct statement(s) is (are)

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Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 31

If

n^{2} \sin^{2} x - 2 \sin x - (2 n + 1) = 0, x \in [0, \frac{π}{2}]

has a real root, then positive integral value(s) of

n

can be

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Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 32

Let

x_{1} < α < β < γ < x_{4}, x_{1} < x_{2} < x_{3}

. If

f (x)

is a cubic polynomial with real coefficients such that

(f (α))^{2} + (f (β))^{2} + (f (γ))^{2} = 0, f (x_{1}) f (x_{2}) < 0, f (x_{2}) f (x_{3}) < 0

and

f (x_{1}) f (x_{3}) > 0

, then which of the following are CORRECT?

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Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 33

Let

f (x) = \{\begin{matrix} x^{n} \sin (\frac{1}{x^{2}}) & , x \neq 0 \\ 0 & , x = 0 \end{matrix}, (n \in I)

, then

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Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 34

Suppose that

f

is a differentiable function with the property that

f (x + y) = f (x) + f (y) + x y

and

\lim_{h \to 0} \frac{1}{h} f (h) = 3

(where

[.]

represents greatest integer function), then

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Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 35

If

f (x) = x^{2} - 2 |x|

and

g (x) = \{\begin{matrix} \min . {f (t) : - 2 \leq t \leq x, - 2 \leq x \leq 0 \\ \max . {f (t) : 0 \leq t \leq x, 0 \leq x \leq 3 \end{matrix}\}

. Then

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Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 36

If

f (x) = \{\begin{matrix} \frac{x \cdot l n (c o s x)}{l n (1 + x^{2})}, & x \neq 0 \\ 0, & x = 0 \end{matrix}

, then

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Mathematics Crash Course JEE Advanced>Chapter 19 - Continuity and Differentiability>Exercise 1>Q 37

Let $f (x)$ be a function defined on $(- a, a)$ with $a > 0$ . Assume that $f (x)$ is continuous at $x = 0$ and $\lim_{x \to 0} \frac{f (x) - f (k x)}{x} = α$ ,
where $k \in (0, 1)$ , then