If $f\left( x \right) = \frac{1}{{x + 1}} - \log \left( {1 + x} \right),\;x > 0$, then $f$ is

both increasing and decreasing function

If $f\left( x \right) = 3{x^4} + 4{x^3} - 12{x^2} + 12$, then $f\left( x \right)$ is

Increasing in $\left( { - 2,{\rm{ }}0} \right)$ and in $\left( {1,\;\infty } \right)$

Increasing in $\left( { - \infty ,\; - 2} \right)$ and in (0, 1)

Decreasing in $\left( { - 2,{\rm{ }}0} \right)$ and in (0, 1)

Decreasing in $\left( { - \infty ,\; - 2} \right)$ and in $\left( {1,\;\infty } \right)$

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JEE • MATHEMATICS • INCREASING & DECREASING

Increasing & decreasing practice · Medium

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 92 questions

1 Medium

1 of 92

Analyzing +4 | -1

If the function

f\left( x \right) = \frac{{K\sin x + 2\cos x}}{{\sin x + \cos x}}

is increasing for all values of

x

, then

View answer and explanation

Correct option: D

K > 2

Option D (

K > 2

) is correct because for the function to be increasing, its derivative must be greater than 0, leading to the condition K > 2.

2 Medium

2 of 92

Analyzing +4 | -1

f\left( x \right) = {x^3} + b{x^2} + cx + d

and

0 < {b^2} < c

, then in

\left( { - \infty ,\;\infty } \right)

View answer and explanation

Correct option: A

f\left( x \right)

is strictly increasing function

Option A (

f(x)

is strictly increasing function) is correct because the derivative of the function is always positive, indicating that the function is strictly increasing.

3 Medium

3 of 92

Analyzing +4 | -1

f\left( x \right) = \left( {\frac{{{e^{2x}} - 1}}{{{e^{2x}} + 1}}} \right)

View answer and explanation

Correct option: A

An increasing

Option A (An increasing) is correct because the derivative of the function is positive for all x in R, indicating an increasing function.

4 Medium

4 of 92

Analyzing +4 | -1

The function

f\left( x \right) = \cos \left( {\pi /x} \right)

is increasing in the interval

View answer and explanation

Correct option: D

None of these

None of the provided options are correct. The function is increasing in the interval

(\frac{1}{2n+1}, \frac{1}{2n})

for

n \in N

5 Medium

5 of 92

Analyzing +4 | -1

\frac{{{x^2}}}{{f\left( {4a} \right)}} + \frac{{{y^2}}}{{f\left( {{a^2} - 5} \right)}}

represents an ellipse with major axis as

y

-axis and

f

is a decreasing function, then

View answer and explanation

Correct option: D

a \in \left( { - 1,5} \right)

Option D (

a \in (-1,5)

) is correct because it satisfies the condition for the given equation to represent an ellipse with the major axis as the y-axis, considering that f is a decreasing function.

6 Medium

6 of 92

Analyzing +4 | -1

f:[1,10] \rightarrow[1,10]

is a non-decreasing function and

g:[1,10] \rightarrow[1,10]

is a non-increasing function. Let

h(x)=f(g(x))

with

h(1)=1

. Then,

h(2)

View answer and explanation

Correct option: C

is equal to one

Option C (is equal to one) is correct because since f is non-decreasing and g is non-increasing, h(x) = f(g(x)) is non-increasing and h(1) = 1, implying h(x) is a constant function equal to 1.

7 Medium

7 of 92

Analyzing +4 | -1

The values of '

x

' for which the function

\left( {a + 2} \right){x^3} - 3\;a\;{x^2} + 9ax - 1

decreases monotonically throughout for all real

x

are

View answer and explanation

Correct option: D

- \infty < a \le - 3

Option D (

- \infty < a \le - 3

) is correct because it ensures that the derivative of the given function is always non-positive, making the function monotonically decreasing.

8 Medium

8 of 92

Analyzing +4 | -1

Let

f(x)=2 \sin ^3 x-3 \sin ^2 x+12 \sin x+5

0 \leq x \leq \frac{\pi}{2}

. Then,

f(x)

, is

View answer and explanation

Correct option: B

Increasing on

[0, \pi / 2]

Option B (Increasing on

[0, \pi / 2]

) is correct because the derivative of

f(x)

is positive on the interval

[0, \pi / 2]

, indicating that the function is increasing.

9 Medium

9 of 92

Analyzing +4 | -1

Let

f(x)

be a function given by

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

for all

x \in[1,4),

where [.] denotes the greatest integer function. Then,

f(x)

is monotonically

View answer and explanation

Correct option: C

Increasing on

[1,2)

Option C (Increasing on

[1,2)

) is correct because the derivative of

f(x)

is positive on this interval, indicating that

f(x)

is increasing.

10 Medium

10 of 92

Analyzing +4 | -1

Consider the following two propositions

P_1

and

P_2

\mathbf{P}_1

: If

f

and

g

are increasing functions on

[a, b]

, then

f+g

is also increasing on

[a, b]

\mathbf{P}_2

: If

f(x)=x

and

g(x)=1-x

are defined on

[0,1]

, then the product

f g

is increasing on

[0,1]

.
Then

View answer and explanation

Correct option: B

\mathrm{P}_1

is true, but

\mathrm{P}_2

is false

Option B is correct because P1 is true (sum of increasing functions is increasing), while P2 is false (product fg is not increasing on [0,1]).

11 Medium

11 of 92

Analyzing +4 | -1

The function

f\left( x \right) = 1 - {x^3} - {x^5}

is decreasing for

View answer and explanation

Correct option: D

All values of

x

Option D (All values of

x

) is correct because the derivative of the function is always negative, indicating that the function is always decreasing.

12 Medium

12 of 92

Analyzing +4 | -1

The function

f(x)=3 \cos ^4 x+10 \cos ^3 x+6 \cos ^2 x-3

0 \leq x \leq \pi

View answer and explanation

Correct option: B

increasing in

\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right)

Option B is correct because the first derivative of the function is positive in the interval (

\frac{{\pi}}{{2}}

\frac{{2 \pi}}{{3}}

), indicating that the function is increasing in that interval.

13 Medium

13 of 92

Analyzing +4 | -1

y = {\left\{ {x\left( {x - 3} \right)} \right\}^2}

increases for all values of

x

lying in the interval

View answer and explanation

Correct option: A

0 < x < \frac{3}{2}

Option A (

0 < x < \frac{3}{2}

) is correct because the derivative of the function is positive in this interval, indicating that the function is increasing.

14 Medium

14 of 92

Application +4 | -1

The longest distance of the point

\left( {a,\;0} \right)

from the curve

2{x^2} + {y^2} - 2\;x = 0

, is given by

View answer and explanation

Correct option: D

\sqrt {1 - 2\;a + 2\;{a^2}}

Option D (

\sqrt {1 - 2\;a + 2\;{a^2}}

) is correct because it represents the maximum distance calculated by finding the critical point of the distance function.

15 Medium

15 of 92

Analyzing +4 | -1

Let

f(x)=(1-x)^2 \sin ^2 x+x^2

for all

x \in R

, and let

g(x)=\int\limits_{1}^x\left\{\frac{2(t-1)}{t+1}-\log _e t\right\} f(t)\, dt

for all

x \in(1, \infty)

. Then, which of the following is true?

View answer and explanation

Correct option: B

g

is decreasing on

(1, \infty)

The correct answer is that g is decreasing on (1, ∞) because the derivative of g(x) is negative for all x > 1, indicating a decreasing function.

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16 Medium

16 of 92

Analyzing +4 | -1

Let

f

and

g

be two differentiable functions on

R

such that

f^{\prime}(x)>0

and

g^{\prime}(x)<0

for all

x \in R

. Then for all

\mathrm{x}

View answer and explanation

Correct option: B

f(g(x))>f(g(x+1))

Option C (Statement-1 is true; Statement-2 is false) is correct because while the given function is increasing, the reasoning provided in statement-2 is incorrect.

17 Medium

17 of 92

Application +4 | -1

f(x)=a \sin x+x+2

is strictly increasing function then number of possible integral values of '

a

' are

View answer and explanation

Correct option: A

Option A (3) is correct because the condition for f(x) to be strictly increasing leads to the inequality -1 <= a <= 1, resulting in three possible integer values for 'a': -1, 0, and 1.

18 Medium

18 of 92

Analyzing +4 | -1

Let

f(x)=\int e^x(x-1)(x-2) d x

, then

f(x)

decreases in the interval

View answer and explanation

Correct option: C

(1,2)

Option C (

(1,2)

) is correct because

f(x)

decreases when

f'(x) < 0

, which occurs when

e^x(x-1)(x-2) < 0

, leading to the interval

(1,2)

19 Medium

19 of 92

Analyzing +4 | -1

\phi \left( x \right)

is continuous at

x = \alpha

such that

\phi \left( \alpha \right) < 0

and

f\left( x \right)

is a function such that

f'\left( x \right) = \left( {ax - {a^2} - {x^2}} \right)\phi \left( x \right)

for all

x

, then

f\left( x \right)

View answer and explanation

Correct option: A

Increasing in the neighbourhood of

x = \alpha

Option A is correct because the derivative of the function, f'(x), is positive in the neighborhood of x = α, indicating that the function is increasing.

20 Medium

20 of 92

Analyzing +4 | -1

f: R \rightarrow R

defined by

f(x)=3 x+2 a \cos x-5

is invertible, then '

a

' belongs to

View answer and explanation

Correct option: A

[-3 / 2,3 / 2]

The correct answer is A (

[-3 / 2,3 / 2]

) because for

f(x)

to be invertible, its derivative must be either always non-negative or always non-positive, leading to the condition

|a| \leq \frac{3}{2}

21 Medium

21 of 92

Analyzing +4 | -1

f(x)

is a function whose derivative is

e^{-\left(x^2+1\right)^2}(2 x)-e^{-x^4}(2 x)

then

f(x)

increases in

View answer and explanation

Correct option: D

(-\infty, 0)

Option D is correct because the derivative is positive for x < 0, indicating that the function is increasing in the interval (-∞, 0).

22 Medium

22 of 92

Analyzing +4 | -1

f^{\prime}\left(x^2-4 x+3\right) > 0

for all

x \in (2,3)

, then

f(\sin x)

is increasing on

View answer and explanation

Correct option: A

\underset{n \in Z}{\cup} \in\left(2 n \pi,(4 n+1) \frac{\pi}{2}\right)

Option A (

\underset{{n \in Z}}{{\cup}} \in\left(2 n \pi,(4 n+1) \frac{{\pi}}{{2}}\right)

) is correct because

f(\sin x)

is increasing when

f'(\sin x) \cos x > 0

, which occurs on the given interval.

23 Medium

23 of 92

Analyzing +4 | -1

f\left( x \right) = \frac{1}{{x + 1}} - \log \left( {1 + x} \right),\;x > 0

, then

f

View answer and explanation

Correct option: B

a decreasing function

Option D (None of these) is correct because the maximum slope occurs at x = 5, which is at the boundary of the interval.

24 Medium

24 of 92

Analyzing +4 | -1

The set of all

x

for which

\log (1+x) \leq x

, is

View answer and explanation

Correct option: B

(-1, \infty)

Option B (

(-1, \infty)

) is correct because the inequality

\log (1+x) \leq x

holds for all

x

greater than

-1

, which is the domain of the logarithm.

25 Medium

25 of 92

Analyzing +4 | -1

g(x)=2 f\left(2 x^3-3 x^2\right)+f\left(6 x^2-4 x^3-3\right), \forall x \in R

and

f^{\prime \prime}(x)>0, \forall x \in R

, then

g(x)

is increasing in

View answer and explanation

Correct option: B

\left(\frac{-1}{2}, 0\right) \cup(1, \infty)

Option B (

\left(\frac{{-1}}{{2}}, 0\right) \cup(1, \infty)

) is correct because it represents the intervals where the derivative of g(x) is positive, indicating that g(x) is increasing.

26 Medium

26 of 92

Analyzing +4 | -1

The function

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

increases, if

View answer and explanation

Correct option: B

\pi /4 < x < 3\pi /8

Option B (

\pi /4 < x < 3\pi /8

) is correct because the derivative of the function is positive in this interval.

27 Medium

27 of 92

Analyzing +4 | -1

The length of the longest interval, in which the function

3\sin x - 4{\sin ^3}x

is increasing, is

View answer and explanation

Correct option: A

\frac{\pi }{3}

Option A (

\frac{\pi}{3}

) is correct because the function can be simplified to

\sin(3x)

, which increases over an interval of length

\frac{\pi}{3}

28 Medium

28 of 92

Analyzing +4 | -1

f\left( x \right) = 3{x^4} + 4{x^3} - 12{x^2} + 12

, then

f\left( x \right)

View answer and explanation

Correct option: B

Increasing in

\left( { - 2,\mathrm{ }0} \right)

and in

\left( {1,\;\infty } \right)

Option B is correct because the function is increasing where its derivative is positive, which occurs in the intervals (-2, 0) and (1, ∞).

29 Medium

29 of 92

Analyzing +4 | -1

Let

f(x)=x \cos ^{-1}\left(\sin (-|x|), x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right.

Then,

View answer and explanation

Correct option: D

f^{\prime}(x)

is decreasing in

\left(-\frac{\pi}{2}, 0\right)

and increasing in

\left(0, \frac{\pi}{2}\right)

Option D (

f^{\prime}(x)

is decreasing in

\left(-\frac{\pi}{2}, 0\right)

and increasing in

\left(0, \frac{\pi}{2}\right)

) is correct because the second derivative is negative for negative x and positive for positive x.

30 Medium

30 of 92

Analyzing +4 | -1

Let

f(x)=\cos x-1+\left(x^2 / 2\right)

. Then

View answer and explanation

Correct option: D

f

decreases in

(-\infty, 0]

and increases in

[0, \infty)

Option D is correct because the first derivative is negative for x < 0 and positive for x > 0, indicating decreasing and increasing behavior respectively.

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