If the domain of the function f x = log e 2 x + 3 4 x 2 + x - 3 + cos - 1 2 x - 1 x + 2 is ( α , β ] , then the value of 5 β - 4 α is equal to

Step-by-step solution

Given, f x = log e 2 x + 3 4 x 2 + x - 3 + cos - 1 2 x - 1 x + 2 Now, using the definition of log function and cos - 1 function we get, 2 x + 3 4 x 2 + x - 3 > 0 and - 1 ≤ 2 x - 1 x + 2 ≤ 1 ⇒ 2 x + 3 ( 4 x - 3 ) ( x + 1 ) > 0 , 3 x + 1 x + 2 ≥ 0 & x - 3 x + 2 ≤ 0 Now, solving 2 x + 3 ( 4 x - 3 ) ( x + 1 ) > 0 we get, ⇒ x ∈ - 3 2 , - 1 ∪ 3 4 , ∞ . . . . i Similarly for 3 x + 1 x + 2 ≥ 0 we get, x ∈ ( - ∞ , - 2 ) ∪ - 1 3 , ∞ . . . i i And for x - 3 x + 2 ≤ 0 we get, x ∈ ( - 2 , 3 ] . . . i i i Now, taking intersectio