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For real x, the function (x-a)(x-b)/(x-c...

For real x, the function `(x-a)(x-b)/(x-c)` will assume all real values provided

A

`a gt b gt c`

B

`a lt b lt c`

C

`a gt c lt b`

D

`a le c le b`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `y=(x^(2)-(a+b)x+ab)/(x-c)`
`impliesyx-cy=x^(2)-(a+b)x+ab`
`impliesx^(2)-(a+b+y)x+(ab+cy)=0`
For real roots, `D ge 0`
`implies(a+b+y)^(2)-4(ab+cy)ge0`
`implies(a+b)^(2)+y^(2)+2(a+b)y-4ab-4cyge0`
`impliesy^(2)+2(a+b-2x)y+(a-b)^(2)ge0`
which is true for all real values of y.
`therefore" "D le0`
`4(a+b-2a)^(2)-4(a-b)^(2)le0`
`implies4(a+b-2c+a-b)(a+b-2c-a+)le0`
`implies(2a-2c)(2b-c)le0`
`implies(a-c)(b-c)le0`
`implies(c-a)(c-b)le0`
`implies` c mule lie between a and b
`i.e. a le c le b or b le c lea `
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