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Let P (x ) be a quadratic polynomial wi...

Let `P (x ) ` be a quadratic polynomial with real coefficients such that for all real x the relation `2 (1+P (x)) =P (x -1) + P (x +1)` holds.
If `P (0) =8 and P (2) =32` then :
If the range of `P (x)` is `[m ,oo),` then the value of m is :

A

`-12`

B

`-15`

C

`-17`

D

`-5`

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
Let `P(x) = ax^(2) + bx + c`
`P(0) = c rArr c = 8`
`P(2) = 32 rArr 2a + b = 12 rArr a = 1, b = 10`
`P(1) = 19 rArr a + b+ c = 19`
`:. P(x) = x^(2) + 10x + 8 = (x + )^(2) - 17`
`:.` minimum value `= 17 :.` Range `= [-17, oo)`
`rArr m = -17`
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