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Solve tan^(-1)xgtcot^(-1)x graphically. ...

Solve `tan^(-1)xgtcot^(-1)x` graphically. Also find where f(x) = max. `[tan^(-1)x,cot^(-1)x]` is non-differentiable. Also find the range of `y=f(x)`.

Text Solution

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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)`.
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