Question

Let f(x) be a quadratic expression which is positive for all real x. If g(x) = f(x) + f ′(x) + f ′′(x), then for any real x –

• A. g(x) > 0
• B. g(x) ≥ 0
• C. g(x) ≤ 0
• D. g(x) < 0

Answer 1

Answer: (A)

g(x) > 0

Answer 2

Let f(x) = ax² + bx + c > 0

⇒ a > 0, b² – 4ac < 0

g(x) = (ax² + bx + c) + (2ax + b) + 2a

= ax² + (2a + b) x + 2a + b + c

Now a > 0

D = (2a + b)² – 4a (2a + b + c)

= b² – 4ac ; D < 0 ⇒ g(x) > 0