Practicing Success
Let $A = \{x:0 < x <π/2\}$ and f: R→ A be an onto function given by $f(x) = \tan^{-1}(x^2 + x + λ)$, where $λ$ is a constant. Then, |
$λ>0$ $λ≥1/4$ $λ<1/4$ $0≤λ≤1/4$ |
$λ≥1/4$ |
The correct answer is Option (2) → $λ≥1/4$ Since, $f: R→ A$ is an onto function. Therefore, Range of f = A $⇒0≤f(x)≤\frac{π}{2}$ for all $x∈R$ $⇒0 ≤ \tan^{-1}(x^2 + x + λ) ≤π/2$ for all $x∈R$ $⇒0 ≤x^2 + x + λ≤∞$ for all $x∈R$ $⇒x^2 + x + λ≥0$ $⇒1-4λ≤0⇒λ≥\frac{1}{4}$ |