Question

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".

Answer 1

@aum

Answer 2

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

Answer 3

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

Answer 4

give me sec, im doing it

Answer 5

i get X= 1 and X= 7

Answer 6

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?

Answer 7

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.

Answer 8

ok

Answer 9

i get 21

Answer 10

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

Answer 11

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.

Answer 12

so i use 7 and 1? right?

Answer 13

at 7, X= 0

Answer 14

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

Answer 15

ohh ok so lets say 6 and 4

Answer 16

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

Answer 17

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

Answer 18

at 4, x= -27

Answer 19

at 2, x= -15

Answer 20

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

Answer 21

so whats the next step

Answer 22

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.

Answer 23

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

Answer 24

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

Answer 25

what about 3 and 4?

Answer 26

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.

Answer 27

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

Answer 28

i got one more left of this type

Answer 29

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....