Monotonicity

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JEE Main 2013

LEVELJEE Main

The real number $k$ for which the equation, $2 x^{3} + 3 x + k = 0$ has two distinct real roots in $[0, 1]$

lies between -1 and 0

How many real solutions does the equation $x^{7} + 14 x^{5} + 16 x^{3} + 30 x - 560 = 0$ have?

The function $f (x) = tan^{- 1} (sin x + cos x)$ is an increasing function in

(A)

(0, π /2)

(B)

(- π /2, π /2)

(C)

(π /4, π /2)

(D)

(- π /2, π /4)

JEE Main 2005

LEVELJEE Main

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? \\ \textbf{Interval} \quad \textbf{Function} \\ (a) $(- \infty, \infty) x^{3} - 3 x^{2} + 3 x + 3$ \\ (b) $[2, \infty) 2 x^{3} - 3 x^{2} - 12 x + 6$ \\ (c) $(- \infty, 1/3] 3 x^{2} - 2 x + 1$ \\ (d) $(- \infty, - 4) x^{3} + 6 x^{2} + 6$

(A)

(- \infty, \infty), x^{3} - 3 x^{2} + 3 x + 3

(B)

[2, \infty), 2 x^{3} - 3 x^{2} - 12 x + 6

(C)

(- \infty, 1/3], 3 x^{2} - 2 x + 1

(D)

(- \infty, - 4), x^{3} + 6 x^{2} + 6

JEE Advanced 2013

LEVELJEE Main

The number of points in $(- \infty, \infty)$ , for which $x^{2} - x sin x - cos x = 0$ , is

(A)

6

(B)

4

(C)

2

(D)

0

JEE Advanced 2013

LEVELJEE Main

Let $f (x) = x sin π x, x > 0$ . Then for all natural numbers $n, f^{'} (x)$ vanishes at

* Multiple Correct Options

(A)

A unique point in the interval (n, n + 1/2)

(B)

A unique point in the interval (n + 1/2, n + 1)

(C)

A unique point in the interval (n, n + 1)

(D)

Two points in the interval (n, n + 1)

JEE Advanced 2013

LEVELJEE Main

Let $f : [1/2, 1] \to R$ be a positive, non-constant and differentiable function such that $f^{'} (x) < 2 f (x)$ and $f (1/2) = 1$ . Then the value of $\int_{1/2}^{1} f (x) d x$ lies in the interval

(A)

(2 e - 1, 2 e)

(B)

(e - 1, 2 e - 1)

(C)

(\frac{e - 1}{2}, e - 1)

(D)

(0, \frac{e - 1}{2})

JEE Advanced 2005

LEVELJEE Main

If $P (x)$ is a polynomial of degree less than or equal to $2$ and $S$ is the set of all such polynomials so that $P (0) = 0, P (1) = 1$ and $P^{'} (x) > 0\forall x \in [0, 1]$ , then

(A)

S = ϕ

(B)

S = a x + (1 - a) x^{2} \forall a \in (0, 2)

(C)

S = a x + (1 - a) x^{2} \forall a \in (0, \infty)

(D)

S = a x + (1 - a) x^{2} \forall a \in (0, 1)

JEE Advanced 2004

LEVELJEE Main

If $f (x) = x^{3} + b x^{2} + c x + d$ and $0 < b^{2} < c$ , then in $(- \infty, \infty)$

(A)

f (x)

is a strictly increasing function

(B)

f (x)

has a local maxima

(C)

f (x)

is a strictly decreasing function

(D)

f (x)

is bounded

JEE Advanced 2003

LEVELJEE Main

If $f (x) = \int_{x^{2}}^{x^{2} + 1} e^{- t^{2}} d t$ , then $f (x)$ increases in

(0, \infty)

(D)

(- \infty, 0)

JEE Advanced 2002

LEVELJEE Main

The length of the longest interval in which the function $3 sin x - 4 sin^{3} x$ is increasing, is

(A)

\frac{π}{3}

(B)

\frac{π}{2}

(C)

\frac{3 π}{2}

(D)

π

JEE Advanced 2001

LEVELJEE Main

If $f (x) = x e^{x (1 - x)}$ , then $f (x)$ is

(A)

increasing on

[- 1/2, 1]

(B)

decreasing on

R

(C)

increasing on

R

(D)

decreasing on

[- 1/2, 1]

JEE Advanced 2000

LEVELJEE Main

Consider the following statments in $S$ and $R$ \\ $S$ : Both $sin x$ and $cos x$ are decreasing functions in the interval $(\frac{π}{2}, π)$ \\ $R$ : If a differentiable function decreases in an interval $(a, b)$ , then its derivative also decreases in $(a, b)$ . \\ Which of the following is true?

(A)

Both

S

and

R

are wrong

(B)

Both

S

and

R

are correct, but

R

is not the correct explanation of

S

(C)

S

is correct and

R

is the correct explanation for

S

(D)

S

is correct and

R

is wrong

JEE Advanced 2000

LEVELJEE Main

Let $f (x) = \int e^{x} (x - 1) (x - 2) d x$ . Then $f$ decreases in the interval

(A)

(- \infty, - 2)

(B)

(- 2, - 1)

(C)

(1, 2)

(D)

(2, + \infty)

JEE Advanced 2000

LEVELJEE Main

For all $x \in (0, 1)$

(A)

e^{x} < 1 + x

(B)

lo g_{e} (1 + x) < x

(C)

sin x > x

(D)

lo g_{e} x > x

JEE Advanced 1999

LEVELJEE Main

The function $f (x) = sin^{4} x + cos^{4} x$ increases if

(A)

0 < x < π /8

(B)

π /4 < x < 3 π /8

(C)

3 π /8 < x < 5 π /8

(D)

5 π /8 < x < 3 π /4

JEE Advanced 1998

LEVELJEE Main

Let $h (x) = f (x) - (f (x))^{2} + (f (x))^{3}$ for every real number $x$ . Then

(A)

h

is increasing whenever

f

is increasing

(B)

h

is increasing whenever

f

is decreasing

(C)

h

is decreasing whenever

f

is increasing

(D)

nothing can be said in general.

JEE Advanced 1997

LEVELJEE Main

If $f (x) = \frac{x}{s i n x}$ and $g (x) = \frac{x}{t a n x}$ , where $0 < x \leq 1$ , then in this interval

(A)

both

f (x)

and

g (x)

are increasing functions

(B)

both

f (x)

and

g (x)

are decreasing functions

(C)

f (x)

is an increasing function

(D)

g (x)

is an increasing function

JEE Advanced 1995

LEVELJEE Main

The function $f (x) = \frac{l n ( π + x )}{l n ( e + x )}$ is

(A)

increasing on

(0, \infty)

(B)

decreasing on

(0, \infty)

(C)

increasing on

(0, π / e)

, decreasing on

(π / e, \infty)

(D)

decreasing on

(0, π / e)

, increasing on

(π / e, \infty)

JEE Advanced 1994

LEVELJEE Main

The function defined by $f (x) = (x + 2) e^{- x}$ is

(A)

decreasing for all

x

(B)

decreasing in

(- \infty, - 1)

and increasing in

(- 1, \infty)

(C)

increasing for all

x

(D)

decreasing in

(- 1, \infty)

and increasing in

(- \infty, - 1)

JEE Advanced 1993

LEVELJEE Main

If $f (x) = {3 x^{2} + 12 x - 1, 37 - x, - 1 \leq x \leq 2 2 < x \leq 3$ then:

* Multiple Correct Options

(A)

f (x)

is increasing on

[- 1, 2]

(B)

f (x)

is continuous on

[- 1, 3]

(C)

f^{'} (2)

does not exist

(D)

f (x)

has the maximum value at

x = 2

JEE Advanced 1987

LEVELJEE Main

The smallest positive root of the equation, $tan x - x = 0$ lies in

(A)

(0, π /2)

(B)

(π /2, π)

(C)

(π, 3 π /2)

(D)

(3 π /2, 2 π)

Let $- 1 \leq p \leq 1$ . Show that the equation $4 x^{3} - 3 x - p = 0$ has a unique root in the interval $[1/2, 1]$ and identify it.

JEE Advanced 1996

LEVELJEE Main

Let $f (x) = {x e^{a x}, x + a x^{2} - x^{3}, x \leq 0 x > 0$ . Where $a$ is a positive constant. Find the interval in which $f^{'} (x)$ is increasing.

JEE Advanced 1987

LEVELJEE Main

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } How many real solutions does the equation x7+14x5+16x3+30x−560=0 have?

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P(x) is a polynomial of degree less than or equal to 2 and S is the set of all such polynomials so that P(0)=0,P(1)=1 and P′(x)>0∀x∈[0,1], then

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=∫ex(x−1)(x−2)dx. Then f decreases in the interval

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For all x∈(0,1)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=sin4x+cos4x increases if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let h(x)=f(x)−(f(x))2+(f(x))3 for every real number x. Then

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=ln(e+x)ln(π+x)​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function defined by f(x)=(x+2)e−x is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)={3x2+12x−1,37−x,​−1≤x≤22<x≤3​ then:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The smallest positive root of the equation, tanx−x=0 lies in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let −1≤p≤1. Show that the equation 4x3−3x−p=0 has a unique root in the interval [1/2,1] and identify it.

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The real number $k$ for which the equation, $2 x^{3} + 3 x + k = 0$ has two distinct real roots in $[0, 1]$

How many real solutions does the equation $x^{7} + 14 x^{5} + 16 x^{3} + 30 x - 560 = 0$ have?

The function $f (x) = tan^{- 1} (sin x + cos x)$ is an increasing function in

The number of points in $(- \infty, \infty)$ , for which $x^{2} - x sin x - cos x = 0$ , is

Let $f (x) = x sin π x, x > 0$ . Then for all natural numbers $n, f^{'} (x)$ vanishes at

Let $f : [1/2, 1] \to R$ be a positive, non-constant and differentiable function such that $f^{'} (x) < 2 f (x)$ and $f (1/2) = 1$ . Then the value of $\int_{1/2}^{1} f (x) d x$ lies in the interval

If $P (x)$ is a polynomial of degree less than or equal to $2$ and $S$ is the set of all such polynomials so that $P (0) = 0, P (1) = 1$ and $P^{'} (x) > 0\forall x \in [0, 1]$ , then

If $f (x) = x^{3} + b x^{2} + c x + d$ and $0 < b^{2} < c$ , then in $(- \infty, \infty)$

If $f (x) = \int_{x^{2}}^{x^{2} + 1} e^{- t^{2}} d t$ , then $f (x)$ increases in

The length of the longest interval in which the function $3 sin x - 4 sin^{3} x$ is increasing, is

If $f (x) = x e^{x (1 - x)}$ , then $f (x)$ is

Let $f (x) = \int e^{x} (x - 1) (x - 2) d x$ . Then $f$ decreases in the interval

For all $x \in (0, 1)$

The function $f (x) = sin^{4} x + cos^{4} x$ increases if

Let $h (x) = f (x) - (f (x))^{2} + (f (x))^{3}$ for every real number $x$ . Then

If $f (x) = \frac{x}{s i n x}$ and $g (x) = \frac{x}{t a n x}$ , where $0 < x \leq 1$ , then in this interval

The function $f (x) = \frac{l n ( π + x )}{l n ( e + x )}$ is

The function defined by $f (x) = (x + 2) e^{- x}$ is

If $f (x) = {3 x^{2} + 12 x - 1, 37 - x, - 1 \leq x \leq 2 2 < x \leq 3$ then:

The smallest positive root of the equation, $tan x - x = 0$ lies in

Let $- 1 \leq p \leq 1$ . Show that the equation $4 x^{3} - 3 x - p = 0$ has a unique root in the interval $[1/2, 1]$ and identify it.

Let $f (x) = {x e^{a x}, x + a x^{2} - x^{3}, x \leq 0 x > 0$ . Where $a$ is a positive constant. Find the interval in which $f^{'} (x)$ is increasing.

The set of all $x$ for which $ln (1 + x) \leq x$ is equal to .........