Solve `tan^(-1) x gt cot^(-1) x`

Explanation

Graphs of `y=tan^(-1)` x and `y=cot^(-1)` x are plotted as shown in the following figure.

For `tan^(-1)xgtcot^(-1)x," the graph of "y=tan^(-1)"x must lie above the graph of "y=cot^(-1)x`
From the figure, this occurs for `x lt 1,` heance the solution to inequality is `(1, oo)` Now f(x) =max. `{tan^(-1)x, cot^(-1)x}`
`={{:(cot^(-1)x,xlt1),(tan^(-1)x, xge1):}`
Clearly, f(x) is non-differentiale at x=1.
From the graph, the range of `y=f(x)" is "[pi/4,pi)`.