Quadratic EquationHard
Question
Let f (x) be a quadratic expression which is positive for all real values of x.If g (x) = f(x) + f′(x) + f′′(x), then for any real x
Options
A.g (x) < 0
B.g (x) > 0
C.g (x) = 0
D.g(x) ≥ 0
Solution
Let f (x) = ax2 + bx + c > 0 for all x ∈ R
⇒ a > 0
and b2 - 4ac < 0 .....(i)
∴ g(x) = f(x) + f′(x) + f′′(x)
⇒ g(x) = ax2 + bx + c + 2ax + b + 2a
⇒ g(x) = ax2 + x(b + 2a) + (c + b + 2a)
whose discriminant
= (b + 2a)2 - 4a(c + b + 2a)
= b2 + 4a2 + 4ab - 4ac - 4ab - 8a2
= b2 - 4a2 - 4ac
= (b2 - 4ac) - 4a2 < 0 [frol Eq. (i)]
∴ g(x) > 0 for all x as and discriminant < 0.
Thus, g(x) > 0 for all x ∈ R.
⇒ a > 0
and b2 - 4ac < 0 .....(i)
∴ g(x) = f(x) + f′(x) + f′′(x)
⇒ g(x) = ax2 + bx + c + 2ax + b + 2a
⇒ g(x) = ax2 + x(b + 2a) + (c + b + 2a)
whose discriminant
= (b + 2a)2 - 4a(c + b + 2a)
= b2 + 4a2 + 4ab - 4ac - 4ab - 8a2
= b2 - 4a2 - 4ac
= (b2 - 4ac) - 4a2 < 0 [frol Eq. (i)]
∴ g(x) > 0 for all x as and discriminant < 0.
Thus, g(x) > 0 for all x ∈ R.
Create a free account to view solution
View Solution FreeMore Quadratic Equation Questions
The value of α for which the sum of the squares of the roots of the equation x2 - (a - 2)x - a - 1 = 0 assume the l...If x be real then the value of will not lie between -...If α and β are roots of 2x2 − 7x + 6 = 0, then the quadratic equation whose roots are is-...The value of m for which one of the roots of x2 − 3x + 2m = 0 is double of one of the roots of x2 − x + m = ...If α and β be the roots of the equation 2x2 + 2 (a + b) x + a2 + b2 = 0, then the equation whose roots are (&#...