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Hi everyone,
I am studying the concavity of the function
![\[
f(x) = \sqrt{1 - x^a}, \quad a \geq 0
\]](//latex.artofproblemsolving.com/e/f/1/ef14689f1894683256f9a0155037788366496929.png)
on the interval
.
I computed the second derivative and found that for
, the function appears to be concave. However, I am uncertain about the behavior at the endpoints.
Does anyone have insights on confirming concavity rigorously for
and understanding the behavior at the endpoints? Any help would be greatly appreciated!
Thanks!
I am studying the concavity of the function
![\[
f(x) = \sqrt{1 - x^a}, \quad a \geq 0
\]](http://latex.artofproblemsolving.com/e/f/1/ef14689f1894683256f9a0155037788366496929.png)
on the interval
![\( x \in [0,1] \)](http://latex.artofproblemsolving.com/7/6/d/76d950c78d04a107e9fcb49b26ab353531c93b92.png)
I computed the second derivative and found that for

Does anyone have insights on confirming concavity rigorously for

Thanks!