Step 1:

(a) \ \

The function is  . \ \

Apply derivative on each side with respect to x .

Product rule of derivatives :

.

Derivative of the logarithmic function: .

Apply the power rule: .

To find the critical numbers of , equate to zero.

The critical points are . 

Step 2: \ \

(b) The critical points are and the test intervals are .

Since logarithm function domain contains only positive values. \ \

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

Thus, The function is increasing on the interval and \ \

The function is decreasing on the interval .

Step 3: \ \

(c) Consider . \ \

Apply derivative on each side with respect to x .

Find : \ \

.

If and , then has local minimum at .

By the above definition has local minimum at . \ \

Step 4: \ \

(d) .

Determination of concavity and inflection points : 

Equate to zero.

.

Thus, the inflection points are split the intervals into and .

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

Thus, the graph is concave up in the interval and 

The graph is concave down in the interval .

Step 5: \ \

(e) Inflection point at : \ \

.

Inflection point is  .

\ \

\ \

\ \

\ \

\ \

Minimum point is  .

Maximum point is .