Let f : ℝ → ℝ be a function defined by f x = log m 2 sin x - cos x + m - 2 , for some m , such that the range of f is 0 , 2 . Then the value of m is _____ .

Step-by-step solution

Given, f : ℝ → ℝ be a function defined by f x = log m   2 sin x - cos x + m - 2 , for some m , Also given the range of f is 0 , 2 , Now we know that, - 2 ≤ sin x - cos x ≤ 2 ⇒ - 2 ≤ 2 sin x - cos x ≤ 2 (Assuming 2 sin x - cos x = k ) ⇒ - 2 ≤ k ≤ 2                 … 1 Now taking function f x = log m k + m - 2 Given, 0 ≤ f x ≤ 2 ⇒ 0 ≤ log m k + m - 2 ≤ 2 ⇒ 1 ≤