$\"\"$

The function is $\"\"$ .

Differentiate $\"\"$ on each side with respect to $\"\"$ .

$\"\"$

Find the critical points.

Equate $\"\"$ to zero:

$\"\"$

$\"\"$ and $\"\"$

The critical points are $\"\"$ and $\"\"$ .

The test intervals are $\"\"$ , $\"\"$ and $\"\"$ .

$\"\"$

First derivative test :

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

The function $\"\"$ has a local minimum at $\"\"$ , because $\"\"$ changes its sign from negative to positive.

$\"\"$

Local minimum is $\"\"$ .

The function $\"\"$ has a local maximum at $\"\"$ , because $\"\"$ changes its sign from positive to negative.

$\"\"$

Local maximum is $\"\"$ .

$\"\"$

Second derivative test :

$\"\"$ .

Differentiate $\"\"$ on each side with respect to $\"\"$ .

$\"\"$

Substitute $\"\"$ in second derivative.

$\"\"$

Since $\"\"$ , curve is concave up.

Therefore local minimum at $\"\"$ .

Local minimum is $\"\"$ .

Substitute $\"\"$ in second derivative.

$\"\"$

Since $\"\"$ , curve is concave down.

Therefore local maximum at $\"\"$ .

Local maximum is $\"\"$ .

$\"\"$

Local maximum is $\"\"$ .

Local minimum is $\"\"$ .