$\"\"$

Concavity test:

(a) If $\"\"$ for all $\"\"$ in $\"\"$ , then the graph of $\"\"$ is concave upward on $\"\"$ .

(b) If $\"\"$ for all $\"\"$ in $\"\"$ , then the graph of $\"\"$ is concave downward on $\"\"$ .

$\"\"$

The function is $\"\"$ .

Differentiate $\"\"$ with respect to $\"\"$ .

$\"\"$

$\"\"$

Differentiate $\"\"$ with respect to $\"\"$ .

$\"\"$

$\"\"$

Determination of inflection points :

Equate $\"\"$ to zero.

$\"\"$

Thus, the inflection points are $\"\"$ .

Consider the test intervals as $\"\"$ and $\"\"$ .

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Thus, the graph is concave up in the interval $\"\"$ and $\"\"$ .

The graph is concave down in the interval $\"\"$ .

$\"\"$

The function $\"\"$ is concave up in the interval $\"\"$ and $\"\"$ .

The function $\"\"$ is concave down in the interval $\"\"$ .