Final answer:
To determine the intervals where the function f(x) = 2x - tan x is concave downward, calculate the second derivative and find where it is negative. This occurs when tan x is positive, within the given domain. The function is concave downward in intervals where tangent is positive and that match the domain constraint.
Step-by-step explanation:
The question asks on what interval the function f(x) = 2x - tan x is concave downward. To determine this, we need to find the second derivative f''(x), because the concavity of a function can be understood from its second derivative. If f''(x) is negative, the function is concave downward.
First, we find the first derivative of f(x): f'(x) = 2 - sec2x. Then, we find the second derivative: f''(x) = -2sec2x tan x. The function is concave downward where f''(x) is negative, which happens when tan x is positive. So, we must look for intervals where tan x has a positive value within the given domain -7/2 < x < 77/2.
To find these intervals, consider the periodic nature of the tangent function, where it is positive in the intervals from (2k-1)π/2 to (2k)π/2, for integers k. By comparing these intervals to the domain of f(x), we would be able to determine on which intervals f(x) is concave downward.