find the open interval(s) on which the function is increasing or decreasing.

Question

Consider the function on the interval (0, 2π ). For the function, (a) find the open interval(s) on which the function is increasing or decreasing, (b) apply the First Derivative Test to identify all relative extrema, and (c) use a graphing utility to confirm your results.

f (x) = sin x + cos x

Answer 1

  Step 1 :

 (a)

Thu function and the interval is .

Differentiate the function with respect to :

Determination of critical points:

The critical points exist when .

Equate to zero:

Step 2 :

Solve in the interval .

General solution of is , where is an integer.

General solution : .

If , .

If , .

If , .

If , .

The solutions are in the interval .

Step 3 :

Solve in the interval .

General solution : .

If , .

If , .

The solutions are in the interval .

Comments

Contd..

Step 4 :

The critical points are and the test intervals are .

Interval	Test Value	Sign of	Conclusion
			Increasing
			Decreasing
			Decreasing
			Increasing
			Increasing

The function is increasing on the intervals , , and .

The function is decreasing on the intervals and .

Solution :

The function is increasing on the intervals , , and .

The function is decreasing on the intervals and .

Answer 2

Step 1 :

(b)

Thu function and the interval is .

The critical points are .

Find the values of at these critical points.

Compare the four values of to find relative maximum and relative minimum.

Relative maximum value is .

Relative minimum value is .

Solution:

Relative maximum is .

Relative minimum is .

Comments

Step 1 :

(c)

Use the graph to confirm the results.

The graph of the function is :

Observe the graph, 

The function is increasing on the intervals , , and and decreasing on the intervals and .

Relative maximum is .

Relative minimum is .

Solution :

The graph of the function is :