The minimum value of the function fx-x-a1-x-a2-x-a3-x-a101- ai- R exists at x-p- Then A- p is mean of the dataset a 1- a 2- a 3- - a 101B- p is median of the dataset a 1- a 2- a 3- - a 101C- p is mode of the dataset a 1- a 2- a 3- - a 101D- None of these

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Byju's Answer

Standard XII

Mathematics

Range

The minimum v...

Question

The minimum value of the function $f (x) = | x - a_{1} | + | x - a_{2} | + | x - a_{3} | + \dots + | x - a_{101} |, a_{i} \in R$ exists at $x = p$ . Then

p

is mean of the dataset

{a_{1}, a_{2}, a_{3}, . . ., a_{101}}

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p

is median of the dataset

{a_{1}, a_{2}, a_{3}, . . ., a_{101}}

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p

is mode of the dataset

{a_{1}, a_{2}, a_{3}, . . ., a_{101}}

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None of these

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Solution

The correct option is B $p$ is median of the dataset ${a_{1}, a_{2}, a_{3}, . . ., a_{101}}$
For $f (x)$ decreases for the middle term(s) and then increases.
So, $p$ is median of the dataset ${a_{1}, a_{2}, a_{3}, . . ., a_{101}}$

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Similar questions

A 60

H z

exciter sets string into a mode as shown in figure. Length of string is 45

c m

. If displacement amplitudes of particles at

x = 7.5

x = 18

x = 27

are

A_{1}, A_{2} & A_{3}

, then

Q. Let

f : R \to R

be the function

f (x) = (x - a_{1}) (x - a_{2}) + (x - a_{2}) (x - a_{3}) + (x - a_{3}) (x - a_{1})

with

a_{1}, a_{2}, a_{3} \in R

. Then

f (x) \geq 0

if and only if

Let $a_{1}, a_{2}, a_{3}, . . .$ be the terms of an AP. If $\frac{a_{1} + a_{2} + a_{3} + . . . + a_{p}}{a_{1} + a_{2} + a_{3} + . . . + a_{q}} = \frac{p^{2}}{q^{2}}, p \neq q,$ find the value of $\frac{a_{6}}{a_{21}}$ .

Q. State true or false:

a_{1}, a_{2}, a_{3} \in + R

, then

\frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}} + \frac{a_{3}}{a_{1}} \geq 3

Q. Let

A_{1}, A_{2},

and

A_{3}

be the regions on

R^{2}

defined by

A_{1} = {(x, y) : x \geq 0, y \geq 0, 2 x + 2 y - x^{2} - y^{2} > 1 > x + y}, A_{2} = {(x, y) : x \geq 0, y \geq 0, x + y > 1 > x^{2} + y^{2}}, A_{3} = {(x, y) : x \geq 0, y \geq 0, x + y > 1 > x^{3} + y^{3}} .

Denote by

| A_{1} |, | A_{2} |,

and

| A_{3} |

the areas of the regions

A_{1}, A_{2}

and

A_{3}

respectively. Then

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The minimum value of the function f(x)=|x−a1|+|x−a2|+|x−a3|+⋯+|x−a101|,ai∈R exists at x=p. Then

The correct option is B p is median of the dataset {a1,a2,a3,...,a101}For f(x) decreases for the middle term(s) and then increases. So, p is median of the dataset {a1,a2,a3,...,a101}

The minimum value of the function $f (x) = | x - a_{1} | + | x - a_{2} | + | x - a_{3} | + \dots + | x - a_{101} |, a_{i} \in R$ exists at $x = p$ . Then

The correct option is B $p$ is median of the dataset ${a_{1}, a_{2}, a_{3}, . . ., a_{101}}$
For $f (x)$ decreases for the middle term(s) and then increases.
So, $p$ is median of the dataset ${a_{1}, a_{2}, a_{3}, . . ., a_{101}}$