The function $\"\"$ and $\"\"$ are concave up over the interval $\"\"$ .

Therefore the functions $\"\"$ and $\"\"$ .

The composite function is $\"\"$ .

$\"\"$

Apply derivative on each side.

$\"\"$

Again apply derivative on each side.

$\"\"$

As $\"\"$ , the second term $\"\"$ .

As $\"\"$ , the first term $\"\"$ depends on $\"\"$ .

$\"\"$ when $\"\"$ is increasing.

$\"\"$ , if and only if $\"\"$ is increasing.

$\"\"$ is a concave up when $\"\"$ is increasing.