You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Range

Let ℝ be th...

Question

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 $x$ .
II. There exists a real number $B$ such that $f (x) \geq B$ for all $x$ .

I is true and II is false

No worries! We‘ve got your back. Try BYJU‘S free classes today!

I is false and II is true

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

I and II both are true

No worries! We‘ve got your back. Try BYJU‘S free classes today!

I and II both are false

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B I is false and II is true
$f (x)$ will be minimum when $\sqrt{x}$ is $0$ .
$\Rightarrow f (x)_{m i n} = \sqrt{| x |} - log (1 + | x |) = 0 - log 1 = 0$
Minimum value of $f (x)$ is $0$ , when $x = 0$
Thus, $f (x) \geq 0$
And maximum value of $f (x)$ is not defined.
Therefore, $B$ exist but $A$ does not exist.

Suggest Corrections

Similar questions

Q. 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 .

II. There exists a real number

B

such that

f (x) \geq B

for all

x

Q. Let

p (x) = x^{2} + a x + b

have two distinct real roots, where

a, b

are real numbers. Define

g (x) = p (x^{3})

for all real numbers

x

. Then which the following statements are true?
I.

g

has exactly two distinct real roots
II.

g

can have more than two distinct real roots
III. There exists a real number

α

such that

g (x) \geq α

for all real

x

Q. Let

f : R \to R

be a function such that for any irrational number

r,

and any real number

x

we have

f (x) = f (x + r)

. Then,

f

Let $f (x) = (1 - x)^{2} s i n^{2} x + x^{2}$ for all $x \in R$ . Consider the statements:
P: There exists some $x \in R$ such that $f (x) + 2 x = 2 (1 + x^{2}) .$
Q:There exists some $x \in R$ such that $2 f (x) + 1 = 2 x (1 + x) .$
Then

Q. Let

x_{k}

be real numbers such that

x_{k} \geq k^{4} + k^{2} + 1

for

1 \leq k \leq 2018

. Denote

N = 2018 \sum k = 1 k

. Consider the following inequalities :-
I.

{(2018 \sum k = 1 k x_{k})}_{k}^{2} \leq N (2018 \sum k = 1 k {x_{k}}^{2})

II.

{(2018 \sum k = 1 k x_{k})}_{k}^{2} \leq N (2018 \sum k = 1 k^{2} {x_{k}}^{2})

Join BYJU'S Learning Program

Let R be the set of real numbers and f:R→R be given by f(x)=√|x|−log(1+|x|). We now make the following assertions:I. There exists a real number A such that f(x)≤A for all x.II. There exists a real number B such that f(x)≥B for all x.

The correct option is B I is false and II is truef(x) will be minimum when √x is 0.⇒f(x)min=√|x|−log(1+|x|)=0−log1=0Minimum value of f(x) is 0, when x=0Thus, f(x)≥0And maximum value of f(x) is not defined.Therefore, B exist but A does not exist.

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 $x$ .
II. There exists a real number $B$ such that $f (x) \geq B$ for all $x$ .