Let f(x) be a quadratic expression which is positive for all ral x and g(x) = f(x) + f'(x) + f''(x), then for any real x,

Options

A. `g(x) lt 0`
B. `g(x) gt 0`
C. `g(x) = 0`
D. `g(x) le 0`

Answer

B

Explanation

Let `f(x) = ax^(2) + bx + c, a gt 0, b^(2) lt 4ac" "( :' f(x) gt 0)`
Now, ` g(x) = ax^(2) + bx + c + 2ax + b + 2a`
`= ax^(2) + (b+2a) x + 2a + b+ c`
Now, `(b+2a)^(2) - 4a ( 2a + b + c)`
`=b^(2) + 4ab + 4a^(2) - 8a^(2) - 4ab-4ac`
`= b^(2) - 4ac - 8a^(2) lt 0 `
`rArr g(x) gt 0 `