Let f : R → R be defined by f x = x 2 - 3 x - 6 x 2 + 2 x + 4 Then which of the following statements is(are) TRUE?

f is increasing in the interval ( 1 , 2 )

MEDIUM

Earn 100

Consider the following statements :

$1 .$ $f (x) = \ln x$ is increasing in $(0, \infty)$

$2 .$ $g (x) = e^{x} + e^{\frac{1}{x}}$ is decreasing in $(0, \infty)$

Which of the statements given above is/are correct?

22.22% studentsanswered this correctly

Important Questions on Application of Derivatives

HARD

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

The number of points, where the curve $y = x^{5} - 20 x^{3} + 50 x + 2$ crosses the $x$ -axis, is $?$

HARD

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

Let $f : R \to R$ be defined by $f (x) = \frac{x^{2} - 3 x - 6}{x^{2} + 2 x + 4}$

Then which of the following statements is(are) TRUE?

MEDIUM

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

The number of real solutions of the equation

27^{\frac{1}{x}} + 12^{\frac{1}{x}} = 2 \times 8^{\frac{1}{x}}

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

a

be an integer such that all the real roots of the polynomial

2 x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 10 x + 10

lie in the interval

(a, a + 1)

. Then,

| a |

is equal to ______.

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 surface area of a sphere when the volume is increasing at the same rate as its radius, 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?

EASY

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

Let $f$ be a real valued function, defined on $R - \{- 1, 1\}$ and given by $f (x) = 3 \log_{e} |\frac{x - 1}{x + 1}| - \frac{2}{x - 1}$ . Then in which of the following intervals, function $f (x)$ is increasing?

MEDIUM

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

Let

f

and

g

be increasing and decreasing functions respectively from

[0, \infty)

[0, \infty)

and

h (x) = f \{g (x)\}

. If

h (0) = 0

, then

h (x) - h (1)

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

EASY

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

m

is the minimum value of

k

for which the function

f (x) = x \sqrt{k x - x^{2}}

is increasing in the interval

[0, 3]

and

M

is the maximum value of

f

[0, 3]

when

k = m,

then the ordered pair

(m, M)

is equal to:

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

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 .

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 $x i n (1, e)$	(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

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

The number of real roots of the equation

e^{6 x} - e^{4 x} - 2 e^{3 x} - 12 e^{2 x} + e^{x} + 1 = 0

is:

EASY

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

The equation

x \log x = 3 - x

HARD

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

Let

f (x) = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!}

. The number of real roots of

f (x) = 0

Consider the following statements : 1. f(x)=ln x is increasing in 0, ∞ 2. g(x)=ex+e1x is decreasing in 0, ∞ Which of the statements given above is/are correct?

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Consider the following statements :

$1 .$ $f (x) = \ln x$ is increasing in $(0, \infty)$

$2 .$ $g (x) = e^{x} + e^{\frac{1}{x}}$ is decreasing in $(0, \infty)$

Which of the statements given above is/are correct?