OpenStudy (anonymous):

Let f(x)=x3−12x2+21x+3. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima). 1. f is increasing on the intervals 2. f is decreasing on the intervals 3. The relative maxima of f occur at x = 4. The relative minima of f occur at x = Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word "none".

3 years ago
OpenStudy (anonymous):

@aum

3 years ago
OpenStudy (anonymous):

the derivative is 3 (x^2-8 x+7)

3 years ago
OpenStudy (aum):

Correct. Equate the derivative, f'(x), to zero and solve for x. Those are called the critical points.

3 years ago
OpenStudy (anonymous):

give me sec, im doing it

3 years ago
OpenStudy (anonymous):

i get X= 1 and X= 7

3 years ago
OpenStudy (aum):

Correct. Consider three intervals: (-inf, 1); (1, 7); (7, inf) Pick a convenient sample point in each interval. For example, 0 would be a convenient number in the interval (-inf, 1). Put x = 0 in f'(x). No need to actuallu evaluate. Just figure out the sign of f'(0). Is it positive? Negative?

3 years ago
OpenStudy (aum):

Since you have two critical points, 1 and 7, there will be three intervals. Left of 1, in between 1 and 7, and right of 7.

3 years ago
OpenStudy (anonymous):

ok

3 years ago
OpenStudy (anonymous):

i get 21

3 years ago
OpenStudy (aum):

In each interval we have to determine if f'(x) is positive, negative or zero.

3 years ago
OpenStudy (aum):

Correct. Like I said before, the actual number is not important-only if it is positive, zero or negative. In (-inf, 1), f'(x) > 0. That means, f(x) is increasing in the interval (-inf, 1). Pick a convenient point in each of the other two intervals, evaluate the sign of f'(x) and conclude if f(x) is increasing or decreasing in those two intervals.

3 years ago
OpenStudy (anonymous):

so i use 7 and 1? right?

3 years ago
OpenStudy (anonymous):

at 7, X= 0

3 years ago
OpenStudy (aum):

Exclude the end points. If you use 7 and 1 you wll get f'(x) is zero.

3 years ago
OpenStudy (anonymous):

ohh ok so lets say 6 and 4

3 years ago
OpenStudy (aum):

Pick a small integer if possible in the interval as a convenient point to evaluate f'(x).

3 years ago
OpenStudy (aum):

In (1,7) I will pick the smallest integer, which will be 2.

3 years ago
OpenStudy (anonymous):

at 4, x= -27

3 years ago
OpenStudy (anonymous):

at 2, x= -15

3 years ago
OpenStudy (aum):

at x = 4, f'(x) = -27 (NOT at 4, x= -27)

3 years ago
OpenStudy (anonymous):

so whats the next step

3 years ago
OpenStudy (aum):

f'(x) is negative in the interval (1,7). That means f(x) is decreasing in the interval (1,7). Next, choose a convenient point in the interval (7, inf). Pick smallest integer if possible. Pick 8. Find f'(8). based on the sign of f'(x) conclude f(x) is increasing or decreasing.

3 years ago
OpenStudy (anonymous):

at X= 8, f'(x)= 21

3 years ago
OpenStudy (aum):

Since f'(x) is positive in the interval (7, inf), f(x) is increasing in the interval (7, inf).

3 years ago
OpenStudy (anonymous):

what about 3 and 4?

3 years ago
OpenStudy (aum):

Conclusion: Intervals: (-inf, 1) (1, 7) (7, inf) f'(x) positive negative positive f(x) increasing decreasing increasing 1. f(x) is increasing on the intervals: (-inf, 1), (7, inf) 2. f(x) is decreasing on the interval: (1,7) 3. Since f(x) increases on (-inf, 1) and decreases on (1,7), f(x) must attain its relative maxima at x = 1. 4. Since f(x) decreases on (1,7) and increases on (7,inf), f(x) must attain its relative minima at x = 7.

3 years ago
OpenStudy (anonymous):

i solved one of the problems while you were explaining to me

3 years ago
OpenStudy (anonymous):

i got one more left of this type

3 years ago
OpenStudy (aum):

I have to log out now. But follow the same principles outlined here. Given f(x), find f'(x). equate it to zero and solve for x. Those are the critical points. Split the number line into various intervals based on the critical points. Pick a convenient number in each interval, evaluate the sign of f'(x). If f'(x) is positive, then f(x) is increasing. If f'(x) is negative, f(x) is decreasing....

3 years ago