If the quadratic polynomial f(x)=(a-3) x2-2 a x+3 a-7 ranges from [-1, ∞) for every x ∈ R, then the value of a lies in

Question

If the quadratic polynomial f(x)=(a-3) x2-2 a x+3 a-7 ranges from [-1, ∞) for every x ∈ R, then the value of a lies in (A) [0,2] (B) [3,5] (C) [4,6) (D) [5,7]

Tardigrade Admin · Accepted Answer

f(x)=(9-3) x2-2 a x+3 a-7 Since range of f(x) ∈[-1, ∞) ∀ x ∈ R Hence range of (a-3) x2-2 a x+3 a-6 ∈[0, ∞) ∴ a-3>0 and D=0 on solving we get a =6 only

Q. If the quadratic polynomial $f (x) = (a - 3) x^{2} - 2 a x + 3 a - 7$ ranges from $[- 1, \infty)$ for every $x \in R$ , then the value of a lies in

$[0, 2]$

$[3, 5]$

$[4, 6)$

$[5, 7]$

$f (x) = (9 - 3) x^{2} - 2 a x + 3 a - 7$
Since range of $f (x) \in [- 1, \infty) \forall x \in R$
Hence range of $(a - 3) x^{2} - 2 a x + 3 a - 6 \in [0, \infty)$
$∴ a - 3 > 0$ and $D = 0$
on solving we get $a = 6$ only

Q. If the quadratic polynomial f(x)=(a−3)x2−2ax+3a−7 ranges from [−1,∞) for every x∈R, then the value of a lies in

[0,2]

[3,5]

[4,6)

[5,7]