Question

Determine the open intervals on which the graph is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) y = 4x − 2 tan x, − π 2 , π 2

Answer 1

The graph is concave upward for the entire interval [-π/2, π/2].

Step-by-step explanation:

To determine the concavity of the graph of the function y = 4x - 2 tan(x) over the interval [-π/2, π/2], we need to find the intervals where the second derivative is positive or negative. The concavity changes at points where the second derivative equals zero or does not exist.

First, find the first and second derivatives of the function: y' = 4 - 2 sec^2(x) and y'' = -4 sec^2(x) tan(x).

Next, solve y'' = 0 to find the critical points. In this case, the equation has no real solutions, which means there are no points where the concavity changes. Therefore, the graph is concave upward for the entire interval [-π/2, π/2].