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 : 0 , 2 → R be a twice differentiable function such that f ' ' x > 0 , for all x ∈ 0 , 2 . If ϕ x = f x + f 2 – x , then ϕ is

decreasing on 0,1 and increasing on ( 1,2 )

increasing on ( 0,1 ) and decreasing on 1,2

The function f x = x 3 - 3 x is

Increasing in - ∞ , - 1 ∪ 1 , ∞ and decreasing in - 1,1

Increasing in 0 , ∞ and decreasing in - ∞ , 0

Decreasing in 0 , ∞ and increasing in - ∞ , 0

Decreasing in - ∞ , - 1 ∪ 1 , ∞ and increasing in - 1,1

Let f x = x cos - 1 ⁡ - sin ⁡ x , x ∈ - π 2 , π 2 , then which of the following is true?

f ' is decreasing in - π 2 , 0 and increasing in 0 , π 2

f ' is increasing in - π 2 , 0 and decreasing in 0 , π 2

f is not differentiable at x = 0

HARD

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If $f (x)$ and $g (x)$ are two positive and increasing functions, then which of the following is $NOT$ always true?

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Important Questions on Application of Derivatives

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

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>Monotonicity

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)

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

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>Monotonicity

Let

f : [0, 2] \to R

be a twice differentiable function such that

f^{''} (x) > 0,

for all

x \in [0, 2] .

ϕ (x) = f (x) + f (2 - x),

then

ϕ

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

Let

f (x) = 2^{10} x + 1

and

g (x) = 3^{10} x - 1 .

(f o g) (x) = x,

then

x

is equal to:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

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

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

Let

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

Then,

f

is an increasing function in the interval:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

The function

f (x) = 4 \sin^{3} x - 6 \sin^{2} x + 12 sinx + 100

is strictly

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

The function

f

defined by

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

is:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

The sum of non - real roots of the polynomial equation

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

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

Let

f (x) = e^{x} - x

and

g (x) = x^{2} - x, \forall x ϵ R

. Then the set of all

x ϵ R

, where the function

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

is increasing, is:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

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 .

EASY

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

The function

f (x) = x^{3} - 3 x

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

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>Monotonicity

Let

f (x) = x \cos^{- 1} (- \sin |x|), x \in [- \frac{π}{2}, \frac{π}{2}],

then which of the following is true?

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

The function

f

given by

f (x) = x^{3} e^{x}

is increasing on the interval

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

The interval in which the function

f (x) = x^{3} - 6 x^{2} + 9 x + 10

is increasing, is

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

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>Monotonicity

The number of points in

(- \infty, \infty)

, for which

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

, is:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

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 .

If fx and gx are two positive and increasing functions, then which of the following is NOT always true?

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

If $f (x)$ and $g (x)$ are two positive and increasing functions, then which of the following is $NOT$ always true?