Consider the function f(x)=1x2+6x+15 a) give the domain of f (in interval notation) equation editorequation editor b) find the critical numbers of f. equation editorequation editor (separate multiple answers by commas.) c) determine the intervals on which f is increasing and decreasing. f is increasing on: equation editorequation editor f is decreasing on: equation editorequation editor d) use the first derivative test to determine whether each critical point is a relative maximum, minimum, or neither. if the function does not have a relative max or min, type none relative maxima occur at x= equation editorequation editor (separate multiple answers by commas.) relative minima occur at x= equation editorequation editor (separate multiple answers by commas.) note: you can earn partial credit on this proble

(a) The domain of the function is

(b) The critical number of the function is

(c) The function is decreasing on the interval and it is increasing on the interval

(d) has a relative minimum point at x = -3

Step-by-step explanation:

We have the following function and we want to find:

(a) The domain of the function is the complete set of possible values of the independent variable.

For this function any real number can be substituted for x and get a meaningful output. Therefore

Domain:

(b) We say that is a critical number of the function if exists and if either of the following are true.

We first need the derivative of the function

the only critical points will be those values of x which make the derivative zero. So, we must solve

(c) To determine the intervals of increase and decrease of the function , perform the following:

Form open intervals with critical number and take a value from every interval and find the sign they have in the derivative.

If f'(x) > 0, f(x) is increasing.

If f'(x) < 0, f(x) is decreasing.

On the interval , take x = -4

f'(x) < 0 therefore f(x) is decreasing

On the interval , take x = 0

f'(x) > 0 therefore f(x) is increasing

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

(d) An extremum point would be a point where is defined and change signs.

decreases (f'(x) < 0) before x = -3, increases after it (f'(x) > 0). So has a relative minimum point at x = -3

We can check our work with the graph of the function.