If `alpha, beta` be the roots of `4x^(8) - 16x + c = 0, c in R` such that `1 lt alpha lt 2 and 2 lt beta lt 3`, then the number of integral values of c is

Answer

3

Explanation

`:'4x^(2)-16x+c=0`
`implies x^(2)-4x+c/4=0`
Let `f(x)=x^(2)-4x+c/4`
Then the following case arises:

Case I `Dgt0`
`=16-cgt0`
`:.clt16`
Case II `f(1)gt0` ltbr `implies1-4+c/4gt0`
`impliesc/4gt3`
`:.cgt12`
Case III `f(2)lt0`
`implies4-8+c/4lt0`
`impliesc/4lt4`
`clt16`
Case IV `f(3)gt0`
`implies9-12+c/4gt0`
`impliesc/4gt3`
`impliescgt12`
Combining all cases, we get
`12ltclt16`
Thus, integral values of c are 13, 14 and 15.
Hence number of integral values of c is 3.