The inequalities y(-1) ≥-4, y(1) ≤ 0 and y(3) ≥ 5 are known to hold for y=a x2+b x+c then the least value of ' a ' is :

Question

The inequalities y(-1) ≥-4, y(1) ≤ 0 and y(3) ≥ 5 are known to hold for y=a x2+b x+c then the least value of ' a ' is : (A) -1 / 4 (B) -1 / 3 (C) 1 / 4 (D) 1 / 8

Tardigrade Admin · Accepted Answer

a-b+c ≥-4 .....(i) and a + b + c ≤ 0 ⇒-a-b-c ≥ 0....(ii) and 9 a +3 b + c ≥ 5 .... (iii) (i) + (ii) ⇒-2 b ≥-4 (iv) ; (ii) + (iii) + (iv) ⇒ 8 a ≥ 1 ⇒ a ≥ 1 / 8

Q. The inequalities $y (- 1) \geq - 4, y (1) \leq 0$ and $y (3) \geq 5$ are known to hold for $y = a x^{2} + b x + c$ then the least value of ' $a$ ' is :

$- 1/4$

$- 1/3$

$1/4$

$1/8$

$a - b + c \geq - 4$ .....(i)
and a + b + c $\leq 0 \Rightarrow - a - b - c \geq 0$ ....(ii)
and $9 a + 3 b + c \geq 5$ .... (iii)
(i) + (ii) $\Rightarrow - 2 b \geq - 4$
(iv) ; (ii) + (iii) + (iv) $\Rightarrow 8 a \geq 1 \Rightarrow a \geq 1/8$

Q. The inequalities y(−1)≥−4,y(1)≤0 and y(3)≥5 are known to hold for y=ax2+bx+c then the least value of ' a ' is :

−1/4

−1/3

1/4

1/8