Solve `sin^(-1)xlecos^(-1)x` graphically. Check the differentiability of f (x) =min. `{sin^(-1)xlecos^(-1)x}`. Also find the range of y = f(x)

Explanation

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

For `sin^(-1)xlecos^(-1)x` the graph of `y=sin^(-1)` x must lie below the graph of `y=cos^(-1)x`, From the figure, this occurs for `-1lexlt1/sqrt2` hence the solution to the inequality is `[-1,1/sqrt2]`.
Now f(x) = min. `{sin^(-1)x,cos^(-1)x}`
`={{:(sin^(-1)x,,-1lexlt1/sqrt2),(cos^(-1)x,,1/sqrt2lexle1):}`
C,er,uy f(x) is non-differentiable at `x=1/sqrt2`.
From the graph, the range of `t=f(x)" is "=[{:(-pi/2,pi/4):}]`