Question

Find all absolute extrema of the given function on each indicated interval.

f(x) = x^3 - 3x + 1 on
a) [0,2]
b) [-3,2]

Answer 1

ok so i substitute them in?

Answer 2

You want to take the derivative and find critical points. Are you able to do that portion onegirl? c:

Answer 3

yes i know how to find the critical points

Answer 4

or not guess this is something else

Answer 5

Since we're given an interval, we have to also check the `end points`.

So if you've found the critical points correctly.
Plug them into the function separately, then also plug in the end points (0 and 2 for part a).

Then simply compare all of your values that they produce. The largest will be your max, smallest your min.

Answer 6

f(0) = ?
f(-1) = ?
f(1) = ?
f(2) = ?

Answer 7

okay so the critical points are +/- 1

Answer 8

ok sounds good :)

Answer 9

so now i plug them into the problem right?

Answer 10

Yes, into the original function (what it looked like before you took a derivative).

Answer 11

okay hold on

Answer 12

(1)^3 - 3(1) + 1 = -1

Answer 13

so -1 and -3

Answer 14

f(1) = -1
f(-1) = -3

Ok good. We also need to check the end points now. The ends of our interval.
Plug 0 and 2 into the function.

f(0) = ?
f(2) = ?

Answer 15

Woops I think f(-1) should be giving us +3, not -3.

Answer 16

yes you're right its positive 3, so for f(0) = -2 and f(2) its 3

Answer 17

f(0) = 0^3 - 3(0) + 1
f(0) = 1

Answer 18

ok

Answer 19

so -1, 3, 1 and 3

Answer 20

Ok so we now have the function value at our critical points and at the ends of our interval.

Our interval was from 0 to 2.

Hmmm -1 is NOT in that interval is it?

Answer 21

no

Answer 22

So we'll throw that point out.
These are the points we care about.

f(0) = 1
f(1) = -1
f(2) = 3

Which one is the largest? That will be our maximum.

Answer 23

ok 3 is the largest

Answer 24

Ok good! We could write that like this perhaps, as a coordinate pair.

Max: (2,3)

How about your minimum? :)

Answer 25

the minimum is -1

Answer 26

Min : (1,-1)

Answer 27

Yes very good.
When they ask for `critical points` they simply want the X value.
When they ask for `max` or `min points` they want it as an ordered pair usually.

Answer 28

ok thanks