The function $f(x)=\frac{4 x^3-3 x^2}{6}-2 \sin x+(2 x-1) \cos x$ is

Increases in $(-\infty, 1 / 2)$

Decreases in $(-\infty, 1 / 2)$

Back to INCREASING & DECREASING

Your Progress

0 / 200 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • INCREASING & DECREASING

Increasing & decreasing practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

Practice format and chapter quiz

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.

Take a chapter quiz →

Showing 121-150 of 200 questions

Preparing practice and saved progress…

121 Medium

121 of 200

Analyzing +4 | -1

f(x)=\int\limits_{{x^2}}^{x^2+1} e^{-t^2} d t

, then

f(x)

increases in

View answer and explanation

Correct option: D

(-\infty, 0)

Option D

((-\infty, 0))

is correct because the derivative

f'(x)

is positive only when

x < 0

, indicating that

f(x)

is increasing in this interval.

122 Medium

122 of 200

Analyzing +4 | -1

a, b

c

be real, then

f(x)=\left|\begin{array}{ccc}x+a^2 & a b & a c \\ a b & x+b^2 & b c \\ a c & b c & x+c^2\end{array}\right|

, is decreasing on

View answer and explanation

Correct option: A

\left(-\frac{2}{3}\left(a^2+b^2+c^2\right), 0\right)

Option A (

\left(-\frac{{2}}{{3}}\left(a^2+b^2+c^2\right), 0\right)

) is correct because the function

f(x)

is decreasing when its derivative is negative, which occurs on the given interval.

123 Easy

123 of 200

Application +4 | -1

Given that

f^{\prime}(x) > g^{\prime}(x)

for all

x \in R

and

f(0)=g(0)

, then

View answer and explanation

Correct option: A

f(x) > g(x)

for all

x \in(0, \infty)

and

f(x) < g(x)

for all

x \in(-\infty, 0)

Option A (f(x) > g(x) for all x in (0, infinity) and f(x) < g(x) for all x in (-infinity, 0)) is correct because h(x) = f(x) - g(x) is increasing and h(0) = 0.

124 Medium

124 of 200

Analyzing +4 | -1

The function

f(x)=\frac{4 x^3-3 x^2}{6}-2 \sin x+(2 x-1) \cos x

View answer and explanation

Correct option: A

Increases in

(1 / 2, \infty)

Option A is correct. The function increases in

(1 / 2, \infty)

because its derivative is positive in that interval.

125 Medium

125 of 200

Analyzing +4 | -1

f(x)=\left(a b+b^2+1\right) x+\int\limits_{0}^x\left(\cos ^4 \theta+\sin ^4 \theta\right) d \theta

is an increasing function of

x

for all

x \in R

and

b \in R, b

being independent of

x

, then

View answer and explanation

Correct option: B

a \in(-\sqrt{6}, \sqrt{6})

Option B (

a \in(-\sqrt{6}, \sqrt{6})

) is correct because it ensures that the derivative of

f(x)

is always positive, making

f(x)

an increasing function.

126 Medium

126 of 200

Analyzing +4 | -1

The function

f\left( x \right) = {x^{1/x}}

is increasing in the interval

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the function

f(x) = {x^{1/x}}

is increasing in the interval (0, e), which is not listed in the options.

127 Easy

127 of 200

Analyzing +4 | -1

If the function

f(x)=3 \cos |x|-6ax+b

increases for all

x \in R

, then the range of values of

a

is given by

View answer and explanation

Correct option: B

(-\infty,-1 / 2)

Option B ((-∞,-1 / 2)) is correct because it ensures that the derivative of

f(x)

is always positive, making

f(x)

an increasing function for all

x \in R

128 Easy

128 of 200

Analyzing +4 | -1

Let

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

be a decreasing function on the set

R

. Then,

a

and

b

satisfy

View answer and explanation

Correct option: B

a^2+3 b+15 < 0

Option B (

a^2+3 b+15 < 0

) is correct because for the function to be decreasing, its derivative must be negative, leading to the given inequality.

129 Easy

129 of 200

Application +4 | -1

The function

f

defined by

f\left( x \right) = {x^3} - 6{x^2} - 36x + 7

is increasing, if

View answer and explanation

Correct option: D

x < - 2

and also

x > 6

Option D (

x < - 2

and also

x > 6

) is correct because it identifies the intervals where the derivative of the function is positive, indicating that the function is increasing.

130 Easy

130 of 200

Analyzing +4 | -1

For all

x \in(0,1)

View answer and explanation

Correct option: B

\log _e(1+x) < x

Option B (

\log _e(1+x) < x

) is correct because the derivative of

g(x)=\log _e(1+x)-x

is negative for

x \in (0,1)

, implying

g(x) < 0

131 Easy

131 of 200

Analyzing +4 | -1

For the function

f(x)=x \cos \frac{1}{x}, x \geq 1

, which one of the following is incorrect?

View answer and explanation

Correct option: A

For at least one

x

in the interval

[1, \infty), f(x+2)-f(x) < 2

The correct answer is A (For at least one

x

in the interval

[1, \infty), f(x+2)-f(x) < 2

) because

f(x+2) - f(x) > 2

for all

x

in the interval, making the statement that it's less than 2 for at least one x incorrect.

132 Easy

132 of 200

Analyzing +4 | -1

f(x)

and

g(x)

are two positive and increasing functions, then

View answer and explanation

Correct option: D

f(x)>1

, then

(f(x))^{g(x)}

is increasing

Option D is correct because if

f(x) > 1

, then

\log(f(x)) > 0

, and since

f(x)

and

g(x)

are positive and increasing,

(f(x))^{g(x)}

is also increasing.

133 Medium

133 of 200

Analyzing +4 | -1

f(x)=x+\sin x, g(x)=e^{-x}, u=\sqrt{c+1}-\sqrt{c}

v=\sqrt{c}-\sqrt{c-1},(c > 1)

, then

View answer and explanation

Correct option: C

g o f(u) > g \circ f(v)

Option C (g o f(u) > g o f(v)) is correct because f is increasing, g is decreasing, and u < v, implying f(u) < f(v) and thus g(f(u)) > g(f(v)).

134 Easy

134 of 200

Analyzing +4 | -1

The set of values of

a

for which the function

f(x)=\left(\frac{a^2-1}{3}\right) x^3+(a-1) x^2+2 x+1

increases on

R

, is

View answer and explanation

Correct option: B

R-[-3,1]

Option B (

R-[-3,1]

) is correct because it represents the set of values of

a

for which the derivative of

f(x)

is always positive, ensuring that

f(x)

is increasing on

R

135 Medium

135 of 200

Analyzing +4 | -1

Suppose

f(x)

is a real-valued differentiable function such that

f(x) f^{\prime}(x)<0

for all real

x

. Then

View answer and explanation

Correct option: D

|f(x)|

is a decreasing function

Option D is correct because the condition f(x)f'(x) < 0 implies that |f(x)| is decreasing, as the square of the function is decreasing.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

136 Medium

136 of 200

Application +4 | -1

The number of integral values of a for which

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

is monotonic in

\forall x \in R

View answer and explanation

Correct option: B

Option B (4) is correct because the function is monotonic when its derivative has a discriminant less than or equal to zero, leading to a range of [1, 4] for 'a', which includes 4 integers.

137 Easy

137 of 200

Analyzing +4 | -1

Let

f(x)=x e^{x(1-x)}, x \in \mathbb{R}

Then

View answer and explanation

Correct option: A

f^{\prime}(x) \geq 0

\left[\frac{-1}{2}, 1\right]

Option A (

f^{\prime}(x) \geq 0

[\frac{{-1}}{{2}}, 1]

) is correct because the derivative is non-negative in the given interval, as shown by the sign analysis of

f'(x)

138 Medium

138 of 200

Analyzing +4 | -1

f(x)

is a non-zero polynomial of degree four, having local extreme points at

x = - 1,\,0,\,1,

then the set

S = \left\{ {x \in R:f(x) = f(0)} \right\}

contains exactly

View answer and explanation

Correct option: D

two irrational and one rational number

The correct answer is D (two irrational and one rational number) because the polynomial equation

f(x) = f(0)

has solutions

x = 0, \pm \sqrt{2}

, resulting in one rational and two irrational roots.

139 Medium

139 of 200

Analyzing +4 | -1

Statement-1: The function

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

is increasing for all

x>0

.
Statement-2: The functions

x^2 e^x

and

x^2 e^{-x}

are increasing for all

x>0

and the sum of two increasing functions in any interval

(a, b)

is an increasing function in

(a, b)

View answer and explanation

Correct option: C

Statement-1 is true; Statement-2 is false.

Option C (-1/18) is correct because the integral can be simplified using trigonometric identities and then evaluated.

140 Medium

140 of 200

Application +4 | -1

The value of

a

in order that

f(x)=\sqrt{3} \sin x-\cos x-2 a x+b

decreases for all real value of

x

, is given by

View answer and explanation

Correct option: B

a \geq 1

Option B (

a \geq 1

) is correct because it ensures that the derivative of the function is always negative, making the function decreasing for all real values of x.

141 Easy

141 of 200

Analyzing +4 | -1

The function

f(x)=\tan x-4 x

is strictly decreasing on

View answer and explanation

Correct option: A

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

Option A ((-π/3, π/3)) is correct because the derivative of the function is negative within this interval, indicating a strictly decreasing function.

142 Easy

142 of 200

Analyzing +4 | -1

The function

f\left( x \right) = \frac{x}{{1 + \left| x \right|}}

View answer and explanation

Correct option: A

Strictly increasing

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

143 Easy

143 of 200

Analyzing +4 | -1

f(x)=\left\{\begin{array}{ll}3 x^2+12 x-1, & -1 \leq x \leq 2 \\ 37-x, & 2 < x \leq 3\end{array}\right.

, then

View answer and explanation

Correct option: A

f(x)

is increasing on

[-1,2]

Option A (

f(x)

is increasing on

[-1,2]

) is correct because the derivative of

f(x)

is positive on this interval, indicating that

f(x)

is increasing.

144 Easy

144 of 200

Analyzing +4 | -1

Select the correct statement from (a), (b), (c), (d) The function

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

View answer and explanation

Correct option: D

Strictly decreasing in the interval

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

Option D is correct because the derivative of the function is negative in the interval (1, ∞), indicating that the function is strictly decreasing.

145 Easy

145 of 200

Application +4 | -1

If the function

f: R \rightarrow R

given by

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

is invertible, then

View answer and explanation

Correct option: B

a \in(-3,3)

Option B (a in (-3,3)) is correct because for the function to be invertible, its derivative must be strictly positive or strictly negative, leading to the condition -3 < a < 3.

146 Hard

146 of 200

Analyzing +4 | -1

In the interval [0, 1], the function

{x^2} - x + 1

View answer and explanation

Correct option: C

Neither increasing nor decreasing

Option C (Neither increasing nor decreasing) is correct because the derivative of the function changes sign within the interval [0, 1].

147 Easy

147 of 200

Analyzing +4 | -1

Let

f^{\prime}(x) > 0

and

g^{\prime}(x) < 0

for all

x \in R

. Then,

View answer and explanation

Correct option: A

f\{g(x)\} > f\{g(x+1)\}

Option A (

f\{g(x)\} > f\{g(x+1)\}

) is correct because

f(x)

is an increasing function and

g(x)

is a decreasing function.

148 Medium

148 of 200

Analyzing +4 | -1

Let

f^{\prime}(\sin x) < 0

and

f^{\prime \prime}(\sin x) > 0

for all

x \in(0, \pi / 2)

and

g(x)=f(\sin x)+f(\cos x)

, then

g(x)

is decreasing in

View answer and explanation

Correct option: B

\left(0, \frac{\pi}{4}\right)

Option A

\left(\frac{\pi}{4}, \frac{\pi}{2}\right)

is incorrect. Option B

\left(0, \frac{\pi}{4}\right)

is correct because g'(x) is negative in this interval, indicating that g(x) is decreasing.

149 Medium

149 of 200

Analyzing +4 | -1

For

x > 1

and

y=\log x

which one of the following is not true?

View answer and explanation

Correct option: C

y > x-1

Option C (

y > x-1

) is not true because for

x > 1

\log x < x-1

. The other options are true based on properties of logarithms and inequalities.

150 Easy

150 of 200

Application +4 | -1

f(x)

is a differentiable function satistying

f^{\prime}(x)<2

for all

x \in R, f(1)=2

, then greatest possible integral value of

f(3)

View answer and explanation

Correct option: C

Option C (5) is correct because it represents the greatest possible integral value of f(3) based on the given conditions and the Mean Value Theorem.

Revision notes and key takeaways

Review INCREASING & DECREASING in your textbook or class notes, then use this set to check your understanding.

Showing 121-150 of 200 questions

Your Progress

Increasing & decreasing practice · Mixed difficulty

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice