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Let f (x) be a quadratic expressinon whi...

Let f (x) be a quadratic expressinon which is positive for all real values of x, If `g(x)=f(x)+f'(x)+f''(x),` then for any real x

A

`g(x) lt 0`

B

`g(x) gt 0`

C

`g (x) =0`

D

`g(x) ge 0`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `f(x)=ax^(2)+bx+c gt0, AAx inR`
`implies a gt0`
`and b^(2)-4aclt0" "...(i)`
`thereforeg(x)=f(x)+f'(x)f''(x)`
`impliesg(x)=ax^(2)+bx+x+2ax+b+2a`
`impliesg(x)=ax^(2)+a(b+2a)+(c+b+2a)`
whose discriminant
`=(b+2a)^(2)-4a(c+b+2a)`
`=b^(2)+4a^(2)+4ab-4ac-4ab-8a^(2)`
`=b^(2)-4a^(2)-4ac=(b^(2)-4ac)-4a^(2)lt0" "["from Eq."(i)]`
`therefore g(x) gt 0AA c, as a gt0` and discriminant `lt0.`
Thus, `g(x)gt0,AAx inR.`
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