Question

Let f(x) =√(9 – x²)  + √(x² – 4). If the range of f is [√a, √b] where a, b ∈N, find the value of (b/a + 3) 

Answer 1

We have f(x) = √(9 – x²)  + √(x² – 4)

Put x = 9cos²θ + 4 sin²θ

Then f(x) = √(5sin²θ) + √(5cos²θ)

= √5 |sinθ| + √5 |cosθ|

= √5(|sinθ| + |cosθ|)

Max value of f(x) = √5

and the min value of f(x) =√5(1/√2 + 1√2)

= √5 ×√2 = √10

Thus, R_f = [√5, √10]

So, a = 5 and b = 10

Hence, the min value of (b/a + 3)

= (10/5 + 3) = 2 + 3 = 5