Let f ( x )= ax 2+ bx + c ( a

Question

Let f ( x )= ax 2+ bx + c ( a < b ) and f ( x ) ≥ 0 ∀ x ∈ R. Find the minimum value of ( a + b + c / b - a ). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

f (1)= a + b + c f(-2)=4 a-2 b+c text hence f (1)- f (-2)=3( b - a ) E=(a+b+c/b-a)=(3 f(1)/f(1)-f(-2))=(3/1-(f(-2))f(1)) text Hence E min . text occurs when f(-2)=0 text Hence E min .=3

Q. Let $f (x) = a x^{2} + b x + c (a < b)$ and $f (x) \geq 0\forall x \in R$ . Find the minimum value of $\frac{a + b + c}{b - a}$ .

$f (1) = a + b + c$
$f (- 2) = 4 a - 2 b + c$
$hence f (1) - f (- 2) = 3 (b - a)$
$E = \frac{a + b + c}{b - a} = \frac{3 f ( 1 )}{f ( 1 ) - f ( - 2 )} = \frac{3}{1 - \frac{f ( - 2 )}{f ( 1 )}}$
$Hence E_{m i n .} occurs when f (- 2) = 0$
$Hence E_{m i n .} = 3$

Q. Let f(x)=ax2+bx+c(a<b) and f(x)≥0∀x∈R. Find the minimum value of b−aa+b+c​.

f(1)=a+b+c f(−2)=4a−2b+c hence f(1)−f(−2)=3(b−a) E=b−aa+b+c​=f(1)−f(−2)3f(1)​=1−f(1)f(−2)​3​ Hence Emin.​ occurs when f(−2)=0 Hence Emin.​=3

Q. Let $f (x) = a x^{2} + b x + c (a < b)$ and $f (x) \geq 0\forall x \in R$ . Find the minimum value of $\frac{a + b + c}{b - a}$ .

$f (1) = a + b + c$
$f (- 2) = 4 a - 2 b + c$
$hence f (1) - f (- 2) = 3 (b - a)$
$E = \frac{a + b + c}{b - a} = \frac{3 f ( 1 )}{f ( 1 ) - f ( - 2 )} = \frac{3}{1 - \frac{f ( - 2 )}{f ( 1 )}}$
$Hence E_{m i n .} occurs when f (- 2) = 0$
$Hence E_{m i n .} = 3$