The function f x = x + 2 , x < - 1 x 2 , - 1 ≤ x < 1 ( x - 2 ) 2 , x ≥ 1

has two local maxima and two local minima

is such that f ' x change its sign exactly twice

The function f x = x + 2 , x < - 1 x 2 , - 1 ≤ x < 1 ( x - 2 ) 2 , x ≥ 1

has two local maxima and two local minima

A function f is defined by f x = ∫ 0 π c o s t c o s x - t d t , 0 ≤ x ≤ 2 π , then which of the following hold(s) good

There exists atleast one c ∈ 0, 2 π it f ' c = 0

f x is continuous but not differentiable in 0 , 2 π

Let f x = x - 1 x 2 , then which of the following is/are correct?

x = 3 is the point of inflection.

For the question, choose the correct answers from the given options: Statement I: For the function f ( x ) = 15 - x , x < 2 2 x - 3 , x ≥ 2 , x = 2 has neither a maximum nor a minimum point. Statement II: f ' x does not exist at x = 2 .

Statement I is false. Statement II is true.

Statement I is true, Statement II is also true. Statement II is the correct explanation of Statement I.

Statement I is true. Statement II is also true. Statement II is not the correct explanation of Statement I.

Statement I is true. Statement II is false

For the following questions, choose the correct answers from the codes (a), (b), (c) and (d) defined as follows: Statement I: ϕ x = ∫ 0 x 3 sin t + 4 cos t d t , π 6 , π 3 . ϕ x attains its maximum value at x = π 3 . Statement II: ϕ x = ∫ 0 x 3 s i n t + 4 c o s t d t , ϕ x is increasing function in π 6 , π 3 .

Statement I is true. Statement II is also true. Statement II is the correct explanation of Statement I.

Statement I is true. Statement II is also true. Statement II is not the correct explanation of Statement I.

Statement I is true. Statement II is false.

Statement I is false. Statement II is true.

Let u = c + 1 - c and v = c - c - 1 , c > 1 and let f x = l n 1 + x , ∀ x ∈ - 1 , ∞ . Statement I: f u > f v , ∀ c > 1 because Statement II: f x is increasing function, hence for u > v , f ( u ) > f ( v ) .

Statement I is false. Statement II is true

Statement I is true, Statement II is also true, Statement II is the correct explanation of Statement I.

Statement I is true. Statement II is also true. Statement II is not the correct explanation of Statement I.

Statement I is true. Statement II is false

For the following questions, choose the correct answers from the codes (a), (b), (c) and (d) defined as follows: Let f 0 = 0 , f π 2 = 1 , f 3 π 2 = - 1 be a continuous and twice differentiable function. Statement I: f ' x ≤ 1 for atleast one x ∈ 0 , 3 π 2 because Statement II: According to Rolle's theorem, if y = g x is continuous and differentiable ∀ x ∈ a , b and g a = g b , then there exists atleast one c such that g ' c = 0 .

Statement I is true. Statement II is also true. Statement II is the correct explanation of Statement I.

Statement I is true. Statement II is also true. Statement II is not the correct explanation of Statement I.

Statement I is true. Statement II is false

Statement I is false. Statement II is true.

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The function $f (x) = \{\begin{matrix} x + 2 & , & x < - 1 \\ x^{2} & , & - 1 \leq x < 1 \\ (x - 2)^{2} & , & x \geq 1 \end{matrix}$

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Important Questions on Monotonicity, Maxima and Minima

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1>Q 44

A function

f

is defined by

f (x) = \int_{0}^{π} c o s t c o s (x - t) d t, 0 \leq x \leq 2 π

, then which of the following hold(s) good

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1>Q 45

Let

f (x) = \frac{x - 1}{x^{2}}

, then which of the following is/are correct?

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IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1>Q 46

For the following questions, choose the correct answers from the codes (a), (b), (c) and (d) defined as follows:
Statement I: The equation

3 x^{2} + 4 a x + b = 0

has atleast one root in

(0, 1)

, if

3 + 4 a = 0

.
Statement II:

f (x) = 3 x^{2} + 4 x + 6

is continuous and differentiable in

(0, 1)

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JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1>Q 47

For the question, choose the correct answers from the given options:
Statement I: For the function

f (x) = \{\begin{cases} 15 - x, & x < 2 \\ 2 x - 3, & x \geq 2 \end{cases}

x = 2

has neither a maximum nor a minimum point.
Statement II:

f^{'} (x)

does not exist at

x = 2

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1>Q 48

For the following questions, choose the correct answers from the codes (a), (b), (c) and (d) defined as follows:
Statement I:

ϕ (x) = \int_{0}^{x} (3 \sin t + 4 \cos t) d t, [\frac{π}{6}, \frac{π}{3}] . ϕ (x)

attains its maximum value at

x = \frac{π}{3}

.
Statement II:

ϕ (x) = \int_{0}^{x} (3 s i n t + 4 c o s t) d t, ϕ (x)

is increasing function in

[\frac{π}{6}, \frac{π}{3}]

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1>Q 49

Let

f (x)

be a twice differentiable function in

[a, b]

, given that

f (x)

and

f^{"} (x)

has same sign in

[a, b]

.
Statement I:

f^{'} (x) = 0

has at the most one real root in

[a, b]

.
Statement II: An increasing function can intersect the

x

-axis at the most once.

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1>Q 50

Let

u = \sqrt{c + 1} - \sqrt{c}

and

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

and let

f (x) = l n (1 + x), \forall x \in (- 1, ∞)

.
Statement I:

f (u) > f (v), \forall c > 1

because
Statement II:

f (x)

is increasing function, hence for

u > v, f (u) > f (v)

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1>Q 51

For the following questions, choose the correct answers from the codes (a), (b), (c) and (d) defined as follows:
Let $f (0) = 0, f (\frac{π}{2}) = 1, f (\frac{3 π}{2}) = - 1$ be a continuous and twice differentiable function.
Statement I: $|f^{'} (x)| \leq 1$ for atleast one $x \in (0, \frac{3 π}{2})$ because
Statement II: According to Rolle's theorem, if $y = g (x)$ is continuous and differentiable $\forall x \in [a, b]$ and $g (a) = g (b)$ , then there exists atleast one $c$ such that $g^{'} (c) = 0$ .

The function fx=x+2,x<-1x2,-1≤x<1(x-2)2,x≥1

Important Questions on Monotonicity, Maxima and Minima

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The function $f (x) = \{\begin{matrix} x + 2 & , & x < - 1 \\ x^{2} & , & - 1 \leq x < 1 \\ (x - 2)^{2} & , & x \geq 1 \end{matrix}$