Find the intervals of increase or decrease.

Question

(a)  Find the intervals of increase or decrease.

(b)  Find the local maximum and minimum values.

(c)  Find the intervals of concavity and the inflection points.

(d)  Use the information from parts (a)–(c) to sketch the graph.

Check your work with a graphing device if you  have one.

h (x) =  (x + 1)^5 - 5x - 2

Answer 1

Step 1 : 

(a)

Thu function .

Differentiate with respect to x:

Step 2 :

Determination of critical points:

Since  is a polynomial it is continuous for all real numbers.

Thus, the critical points exist when .

Equate to zero:

The critical points are and

Step 3 :

Consider the test intervals as and

Interval	Test Value	Sign of	Conclusion
			Increasing
			Decreasing
			Increasing

Thus, The function is increasing on the intervals and

And The function is decreasing on the interval

Solution :

(a) The function is increasing on the intervals and

     The function is decreasing on the interval

Answer 2

Step 1 :

(b)

Since is changes its sign from positive to negative, h has a local maximum at .

Local maximum is .

Since is changes its sign from negative to positive, h has a local minimum at .

Local minimum is .

Step 2 :

(c)

Differentiate with respect to x:

Determination of concavity and inflection points :

Equate to zero:

Thus, the inflection point is

Comments

Contd........

Step 3 :

Consider the test intervals as and .

Interval	Test Value	Sign of	Concavity
			Down
			Up

Thus, the graph is concave up on the interval

The graph is concave down on the interval .

Inflection point :

Inflection point is

Step 4 : 

(d)

Graph of the function .

 

Solution :

(b) Local maximum is  .

     Local minimum is .

(c) The  function is concave up on the interval

      The  function is concave down on the interval .

(d) Graph of the function .