$\"\"$

(a)

The function is $\"\"$ .

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

$\"\"$

Find the critical points.

Since it is a polynomial it is continuous at all the point.

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

Equate $\"\"$ to zero.

$\"\"$

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

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

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

$\"\"$

(b)

Second derivative test :

$\"\"$ .

$\"\"$

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

$\"\"$

$\"\"$ .

Substitute $\"\"$ in second derivative.

$\"\"$

Since $\"\"$ , curve is neither concave up nor concave down.

Therefore conclusion cannot be made at $\"\"$ .

Substitute $\"\"$ in second derivative.

$\"\"$

Since $\"\"$ , curve is neither concave up nor concave down.

Therefore conclusion cannot be made at $\"\"$ .

Substitute $\"\"$ in second derivative.

$\"\"$

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

Therefore local minimum at $\"\"$ .

$\"\"$

(c)

First derivative test :

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

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

\ \

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

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

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

$\"\"$

Therefore the conclusion cannot be made at $\"\"$ .

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

$\"\"$

Local minimum is $\"\"$ .

The function is incresing from $\"\"$ to $\"\"$ .

Therefore the function has local minimum is $\"\"$ .

$\"\"$

(a) The critical points are $\"\"$ , $\"\"$ and $\"\"$ .

(b) The local minimum at $\"\"$ .