First slide

Introduction to linear inequalities

Question

$If f (x) = a x^{2} + b x + c and f (- 1) \geq - 4, f (1) \leq 0 and f (3) \geq 5, then the least value of a is$

Moderate

A

$\frac{1}{4}$
B

$\frac{1}{8}$
C

$\frac{1}{3}$
D

$- \frac{1}{3}$

Solution

$f (- 1) \geq - 4 - - - - (1)$
$\begin{array}{l} \Rightarrow a - b + c \geq - 4 \\ f (1) \leq 0 \\ \Rightarrow a + b + c \leq 0 \end{array}$
$\Rightarrow - a - b - c \geq 0 - - - (2)$
$f (3) \geq 5$
$and 9 a + 3 b + c \geq 5 - - - - (3)$
$From (i) + (ii) \Rightarrow - 2 b \geq - 4 - - - - (4)$
$From (ii) + (iii) + (iv) \Rightarrow 8 a \geq 1 \Rightarrow a \geq 1 / 8$

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