(a)

Thu function is $\"\"$ , $\"\"$ .

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

$\"\"$

Find the critical points.

Thus, the critical points exist when $\"\"$ .

Equate $\"\"$ to zero:

$\"\"$

The general solution of cosine function is $\"\"$ , where $\"\"$

The solution for $\"\"$ is $\"\"$ , $\"\"$ .

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

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

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

\ \

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

Interval	Test Value	Sign of $\"\"$	Conclusion
$\"\"$	$\"\"$	\ $\"\"$ \	Increasing
$\"\"$	$\"\"$	\ $\"\"$ \	Increasing

The function is increasing over the interval $\"\"$ .

(b)

Find the local maximum and local minimum.

Since the function is increasing, therefore it has neither local maximum nor minimum.

(c)

$\"\"$ .

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

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

The general solution of sine function is $\"\"$ , where $\"\"$ .

The solution for $\"\"$ is $\"\"$ , $\"\"$ .

$\"\"$ , $\"\"$ and $\"\"$ .

The inflection points are at $\"\"$ , $\"\"$ and $\"\"$ .

Substitute $\"\"$ in $\"\"$ $\"\"$ $\"\"$ .

$\"\"$

Substitute $\"\"$ in $\"\"$ $\"\"$ .

$\"\"$

Substitute $\"\"$ in $\"\"$ $\"\"$ .

$\"\"$

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

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

\ Interval \	Test Value	Sign of $\"\"$	Concavity
$\"\"$	$\"\"$	\ $\"\"$ \	Up
$\"\"$	$\"\"$	\ $\"\"$ \	\ Down \
$\"\"$	$\"\"$	\ $\"\"$ \	\ Up \
$\"\"$	$\"\"$	\ $\"\"$ \	Down

The graph is concave up on the intervals $\"\"$ and $\"\"$ .

The graph is concave down on the intervals $\"\"$ and $\"\"$ .

The inflection points are $\"\"$ , $\"\"$ and $\"\"$ .

(d)

Graph :

Graph the function $\"\"$ :

$\"\"$

(a)

The function is increasing over the interval $\"\"$ .

(b)

There is neither local maximum nor local minimum.

(c)

Concave up in the intervals $\"\"$ and $\"\"$ .

Concave down in the intervals $\"\"$ and $\"\"$ .

Inflection points are $\"\"$ , $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

(d)

Graph of the function $\"\"$ is

$\"\"$ .