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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Let $f : R \to R$ be a differentiable function for all values of $x$ and has the property that $f (x)$ and $f^{'} (x)$ have opposite signs for all values of $x$ . Then,

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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?

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

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>Monotonicity

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:

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

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

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

HARD

Mathematics>Differential Calculus>Application of Derivatives>Monotonicity

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

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 .

Let f:R→R be a differentiable function for all values of x and has the property that f(x) and f'(x) have opposite signs for all values of x. Then,

Important Questions on Application of Derivatives

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Let $f : R \to R$ be a differentiable function for all values of $x$ and has the property that $f (x)$ and $f^{'} (x)$ have opposite signs for all values of $x$ . Then,