Let f: R arrow(0, (2 π/3)] defined as f(x)= cot -1(x2-4 x+α). Find the smallest integral value of α such that f ( x ) is into function.

Question

Let f: R arrow(0, (2 π/3)] defined as f(x)= cot -1(x2-4 x+α). Find the smallest integral value of α such that f ( x ) is into function. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Clearly x 2-4 x +α> or ≥ (-1/√3) ∀ x ∈ R ⇒ x 2-4 x +α+(1/√3)>0 ∀ x ∈ R text So, Discriment <0 ⇒ 16-4(α+(1/√3))<0 4-α-(1/√3)<0 ⇒ α>4-(1/√3) ∴ α ∈(4-(1/√3), ∞) text Hence minimum integral α=4

Q. Let f:R→(0,32π​] defined as f(x)=cot−1(x2−4x+α). Find the smallest integral value of α such that f(x) is into function.

Clearly x2−4x+α> or ≥3​−1​∀x∈R ⇒x2−4x+α+3​1​>0∀x∈R So, Discriment <0⇒16−4(α+3​1​)<0 4−α−3​1​<0⇒α>4−3​1​ ∴α∈(4−3​1​,∞) Hence minimum integral α=4

Q. Let $f : R \to (0, \frac{2 π}{3}]$ defined as $f (x) = cot^{- 1} (x^{2} - 4 x + α)$ . Find the smallest integral value of $α$ such that $f (x)$ is into function.