Question

Let f(x) be a polynomial of degree two and

f(x) > 0 for all x∈R if g(x) = f(x) + f ′(x) + f ″(x) then for all x∈R

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

Answer 1

Answer: (B)

g(x) > 0

Answer 2

Q f(x) is polynomial of degree two

∴ f(x) = ax² + bx + c & f(x) > 0 ∀x∈R

∴ a > 0 and D < 0 ⇒ a > 0 and b² – 4ac < 0 ….(i)

Now g(x) = f(x) + f ′(x) + f ″(x)

g(x) = ax² + (b + 2a)x + (2a + c + b)

Now D for g(x) is = (b + 2a)² – 4.a.(2a + c + b)

= b² – 4ac – 4a²

= < 0 Q [b² – 4ac < 0 from(i)]

and coeff. of x² in g(x) = a > 0

∴ g(x) > 0 ∀x∈R