If `x in R ,a n da ,b ,c` are in ascending or descending order of magnitude, show that `(x-a)(x-c)//(x-b)(w h e r ex!=b)` can assume any real value.

Explanation

Let `y = ((x - a) (x - c))/((x - b)) (x ne b)`
or `x^(2) + (-a - c - y) x + ac + by = 0`
Now, x is real . Hence,
`D = (a + c + y)^(2) - 4 (ac + by) ge 0, AA y in` R
or `y^(2) + 2 (a + c - 2b)y + ( a- c)^(2) ge 0, AA y in` R
` rArr D le` 0
`rArr 4(a + c - 2b)^(2) - 4 ( a - c)^(2) le` 0
or `(a + c - 2b + a - c) (a + c - 2b - a + c) le `0
or or `(a + b)(c -b)le` 0 which is ture if ` a lt b lt c or a gt b gt c`