Let F : R → R be a thrice differentiable function. Suppose that F 1 = 0 , F 3 = - 4 and F ′ x < 0 for all x ∈ 1 2 , 3 . Let f x = x F ( x ) for all x ∈ R . The correct statement(s) is(are)

f ′ x ≠ 0 for any x ∈ ( 1 , 3 )

f ′ x = 0 for some x ∈ ( 1 , 3 )

Let f be a twice differentiable function on ℝ such that f 0 = 1 , f 1 = 2 , f ' 0 = - 1 , f ' 1 = 5 and f " x ≥ 0 for x ∈ 0 , 1 . Here, f ' x and f " x denote the first and second order derivative of f at x respectively. Then

f x ≥ 5 x - 3 for x ∈ 0 , 1

Let S be the set of all twice differentiable functions f from ℝ to ℝ such that d 2 f d x 2 x > 0 for all x ∈ - 1 , 1 . For f ∈ S , let X f be the number of points x ∈ - 1 , 1 for which f x = x . Then which of the following statements is(are) true?

There exists a function f ∈ S , such that X f = 2.

There does NOT exist any function f in S such that X f = 1

Let f ( x ) = cos π x , x ≠ 0 , then assuming k as an integer,

f ( x ) decreases in the interval 1 2 k + 2 , 1 2 k + 1

f ( x ) decreases in the interval 1 2 k + 1 , 1 2 k

f ( x ) increases in the interval 1 2 k + 2 , 1 2 k + 1

HARD

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Let $f (x)$ be a derivable function defined over the set of real numbers such that $f (f (x)) = λ (x^{7} + 2 x)$ , $λ \neq 0,$ then $\forall x \in R, f (x)$ is always

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Important Questions on Application of Derivatives

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let

R

be the set of real numbers and

f : R \to R

be given by

f (x) = \sqrt{|x|} - \log (1 + |x|) .

We now make the following assertions:

I .

There exists a real number

A

such that

f (x) \leq A

for all

x

I I .

There exists a real number

B

such that

f (x) \geq B

for all

x .

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let

F : R \to R

be a thrice differentiable function. Suppose that

F (1) = 0, F (3) = - 4

and

F^{'} (x) < 0

for all

x \in (\frac{1}{2}, 3) .

Let

f (x) = x F (x)

for all

x \in R

. The correct statement(s) is(are)

EASY

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

The function

f (x) = \sin x - k x - c

, where

k

and

c

are constants, decreases always when

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

The function

f

defined by

f (x) = x^{3} - 3 x^{2} + 5 x + 7

is:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let $f$ be a twice differentiable function on $ℝ$ such that $f (0) = 1, f (1) = 2, f^{'} (0) = - 1, f^{'} (1) = 5$ and $f^{"} (x) \geq 0$ for $x \in [0, 1]$ .

Here, $f^{'} (x)$ and $f^{"} (x)$ denote the first and second order derivative of $f$ at $x$ respectively. Then

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

The number of points in

(- \infty, \infty)

, for which

x^{2} - x \sin x - \cos x = 0

, is:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

The interval in which the function

f (x) = \frac{4 x^{2} + 1}{x}

is strictly decreasing is

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty)

Column 1 contains information about zeros of

f (x)

f' (x)

and

f'' (x)

Column 2 contains information about the limiting behaviour of

f (x)

f' (x)

and

f'' (x)

at infinity.
Column 3 contains information about increasing-decreasing nature of

f (x)

and

f' (x)

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in $(0, 1)$
(II) $f^{'} (x) = 0$ for some $x in (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in $(0, 1)$
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

Which of the following options is the only CORRECT combination?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

f : R \to R

is a differentiable function such that

f^{'} (x) > 2 f (x)

for all

x \in R

and

f (0) = 1

then

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let S be the set of all twice differentiable functions f from

ℝ

ℝ

such that

\frac{d^{2} f}{d x^{2}} (x) > 0

for all

x \in (- 1, 1)

. For

f \in S

, let

X_{f}

be the number of points

x \in (- 1, 1)

for which

f (x) = x

. Then which of the following statements is(are) true?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

The sum of non - real roots of the polynomial equation

x^{3} + 3 x^{2} + 3 x + 3 = 0

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let

f : ℝ ⟶ ℝ

be a function given by

f (x) = \max \{1 - x, 1 + x, 2\}

. Then

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

f (x) = {(\frac{3}{5})}^{x} + {(\frac{4}{5})}^{x} - 1, x \in R,

then the equation

f (x) = 0

has :

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

The graph of the function $f (x) = x + \frac{1}{8} s i n (2 π x), 0 \leq x \leq 1$ is shown below. Define $f_{1} (x) = f (x), f_{n + 1} (x) = f (f_{n} (x)),$ for $n \geq 1$ .

Question Image

Which of the following statements are true?

$I$ . There are infinitely many $x \in [0,1]$ for which $\underset{n \to \infty}{l i m} f_{n} (x) = 0$

$II$ . There are infinitely many $x \in [0,1]$ for which $\underset{n \to \infty}{l i m} f_{n} (x) = \frac{1}{2}$

$III$ . There are infinitely many $x \in [0,1]$ for which $\underset{n \to \infty}{l i m} f_{n} (x) = 1$

$IV$ . There are infinitely many $x \in [0,1]$ for which $\underset{n \to \infty}{l i m} f_{n} (x)$ does not exist .

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let

f (x) = \sin^{4} x + \cos^{4} x .

Then,

f

is an increasing function in the interval:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty) .

•

Column 1 contains information about zeros of

f (x), f^{'} (x)

and

f^{''} (x) .

•

Column 2 contains information about the limiting behaviour of

f (x), f^{'} (x)

and

f^{''} (x)

at infinity.

•

Column 3 contains information about increasing-decreasing nature of

f (x)

and

f^{'} (x) .

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in (0, 1)
(II) $f^{'} (x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in (0, 1)
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

Which of the following options is the only Incorrect combination?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty) .

•

Column 1 contains information about zeros of

f (x), f^{'} (x)

and

f^{''} (x) .

•

Column 2 contains information about the limiting behaviour of

f (x), f^{'} (x)

and

f^{''} (x)

at infinity.

•

Column 3 contains information about increasing-decreasing nature of

f (x)

and

f^{'} (x) .

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in (0, 1)
(II) $f^{'} (x) = 0$ for some $x i n (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in (0, 1)
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

Which of the following options is the only Correct combination?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let $S = (0, 1) \cup (1, 2) \cup (3, 4)$ and $T = \{0, 1, 2, 3\}$ . Then which of the following statements is(are) true?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let $x_{0} > 0$ . For every natural number $n$ define $s_{n} = \sin (\frac{π x_{0}^{n}}{1 + x_{0}^{2 n}})$ and $c_{n} = \cos (\frac{π x_{0}^{n}}{1 + x_{0}^{2 n}})$ .

Then for all $n$

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Increasing and Decreasing Function

Let

f (x) = \cos (\frac{π}{x}), x \neq 0,

then assuming

k

as an integer,

Let f(x) be a derivable function defined over the set of real numbers such that f(f(x))=λx7+2x), λ≠0, then ∀x∈R, f(x) is always

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Let $f (x)$ be a derivable function defined over the set of real numbers such that $f (f (x)) = λ (x^{7} + 2 x)$ , $λ \neq 0,$ then $\forall x \in R, f (x)$ is always