Find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing

Question

(a) Find the critical numbers of f (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results.

f (x) = - 2x^2 + 4x + 3

Answer 1

Step 1:

(a)

The function is .

Find the critical numbers by equating the first derivative to .

Apply derivative on each side with respect to .

Equate the derivative to .

So the function has critical number at .

Solution :

The function has critical number at .

Answer 2

Step 1:

(b)

The critical point is ,consider a table summarizes the testing of two intervals determined by the critical number.

Test interval
Test value
sign of
Conclusion	Increasing	Decreasing

 So the function is increasing on the interval and decreasing on the interval .

Solution :

The function is increasing on the interval and decreasing on the interval .

Answer 3

Step 1:

(c)

Use first Derivative Test to identify all relative extrema.

The derivative of the function is .

The critical point is ,consider the table summarizes the testing of two intervals determined by the critical number.

Test interval
Test value
sign of
Conclusion	Increasing	Decreasing

 From the fist derivative test the function changing from positive to negative at , then has a relative maximum at .

Find for .

So the function has relative maximum at .

Solution :

The function has relative maximum at .

Answer 4

Step 1:

(d)

Graph the function is .

Now observe the graph :

The function has critical number at .

The function is increasing on the interval and decreasing on the interval .

The function has relative maximum at .

 

Solution :

The function has critical number at .

The function is increasing on the interval and decreasing on the interval .

The function has relative maximum at .