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The minimum value of the fuction f(x)=2...

The minimum value of the fuction `f(x)=2|x-2|+5|x-3|` for all ` x in R , is `

A

3

B

2

C

5

D

7

Text Solution

Verified by Experts

The correct Answer is:
B
We have ltbr gt `f(x)=2|x-2|+5|x-3|`
`= f(x)={{:(" "2(x-2)+5(x-3)=7x-19, x ge 3),(" "2(x-2)-5(x-3)=-3x+11,2le x lt 3 ),(-2(x-2)-5(x-3)=-7x=19, x lt 2):}`

`rArr f(X)={:{(7,x gt 3),(-3,2 lt x lt 3),(-7,x lt 2):}`
Clearly f(x) is increasing for all ` x gt 3 `.It decreases for ` x gt 2` and also for ` x in (2,3)`
`therefore ` Minimum value of `f(x)= f(3)=21 - 19=2`
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