Consider the function f ( x ) = log e ( 1 - x 2 - x ) . Then, which of the following is/are true?

The least value of f does not exist

Consider the function f ( x ) = log e ( 1 - x 2 - x ) . Then, which of the following is/are true?

The least value of f does not exist

If f x = x 3 ( 1 - x ) , x ≤ 0 x log e x + 3 x , x > 0 , then which of the following is not true

f ( x ) has a point of maxima at x = 0 .

f ( x ) has a point of minima at x = e - 4 .

If f x = x , 0 ≤ x ≤ 1 2 - e x - 1 , 1 < x ≤ 2 x - e , 2 < x ≤ 3 and g ' ( x ) = f ( x ) , x ∈ [ 1 , 3 ] , then:

g ( x ) has a local maxima at x = 1 + ln 2 and local minima at x = e

g ( x ) has a local minima at x = 1 + ln 2 and a local maxima at x = e

E M B I B E

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 15

HARD

JEE Main

IMPORTANT

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Consider the function $f (x) = \log_{e} (\sqrt{1 - x^{2}} - x)$ . Then, which of the following is/are true?

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Important Questions on Monotonicity and Maxima-Minima of Functions

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 1

If

f (x) = \{\begin{array}{cl} x^{3} (1 - x), & x \leq 0 \\ x \log_{e} x + 3 x, & x > 0 \end{array}

, then which of the following is not true

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 2

The coordinates of the point on the curve

x^{3} = y (x - a)^{2}

, where the ordinate is minimum is

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 3

The fraction exceeding its

p^{th}

power by the greatest number possible, where

p \geq 2

, is:

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 4

If

f (x) = \{\begin{array}{l} x, & 0 \leq x \leq 1 \\ 2 - e^{x - 1}, & 1 < x \leq 2 \\ x - e, & 2 < x \leq 3 \end{array}\}

and

g^{'} (x) = f (x)

,

x \in [1, 3]

, then:

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 5

If

g (x) = \max (y^{2} - x y)

(where

0 \leq y \leq 1

), then the minimum value of

g (x)

(for real

x

) is:

MEDIUM

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 6

If

a, b \in R

distinct numbers satisfying

|a - 1| + |b - 1| = |a| + |b| = |a + 1| + |b + 1|,

then the minimum value of

|a - b|

is:

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 7

If the equation

2 x^{3} - 6 x + 2 \sin a + 3 = 0

, where

a \in (0

,

π)

has only one real root, then the largest interval in which

a

lies is:

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 6 - Monotonicity and Maxima-Minima of Functions>DPP>Q 8

Let $f$ be a continuous and differentiable function in $(x_{1}, x_{2})$ . If $f (x) . f^{'} (x) \geq x \sqrt{1 - {(f (x))}^{4}}$ , $\lim_{x \to x_{1}^{+}} {(f (x))}^{2} = 1$ and $\lim_{x \to x_{2} -} {(f (x))}^{2} = \frac{1}{2}$ , then the minimum value of $(x_{1}^{2} - x_{2}^{2})$ is: