Question

Numerically estimate the absolute extrema of the given function on the indicated intervals.

f(x) = x^4 - 3x^2 + 2x + 1 on
a) [-1,1]
b) [-3,2]

Answer 1

derivate the function and find x values by equating the derivated equation to zero
then you will have some x values. they are called critical values

Answer 2

yea i already found the critical points

Answer 3

What are they?

Answer 4

x = 1, x = -1.3660, x = 0.36603

Answer 5

Now for the first part of the problem ie a)

Answer 6

Since he said in limits [-1,1], then what are the critical values that lie in the given limit?

Answer 7

so i substitute the critical point into the original problem right?

Answer 8

NOTE : the square brackets are inclusive i mean when you have given that x lies in limit [-2,2]
x may take any value in between -2 and 2, x can also take the values 2 and -2

Answer 9

but the open brackets "( )" are exclusive brackets i mean when you have given that x lies in limit [-2,2]
x may take any value in between -2 and 2, but x can not take the values 2 and -2

Answer 10

got it?

when x lies [-2,2] that means
\[-2\le x \le2\]
but
when x lies in (-2,2) that means
\[-2 < x < 2\]

Answer 11

ok

Answer 12

Now, coming to the problem a)

Answer 13

Now say what are the critical points in the given region [-1,1] out of three you got?

Answer 14

1

Answer 15

and also 0.36603 right ?

Answer 16

cause its in limit [-1,1]

Answer 17

yes

Answer 18

Now while calculating absolute extrema, you should substitute the critical points in given limit and also upper limit and lower limit in the function

Answer 19

So, here for the a) problem
you have to substitute x =1, x= 0.36603, x = -1

Answer 20

ok s (1) ^4 - 3(1)^2 + 2(1) + 1 = 1

Answer 21

and and for the second of i got -3.8480

Answer 22

and for the third i got 1.61592

Answer 23

so what do i do next?

Answer 24

what value did you get for x = 0.36603 ?

Answer 25

i got 1.61592

Answer 26

i got 1.3480

Answer 27

ohh ok

Answer 28

now which values of x has most positive and most negative values?

Answer 29

1.3480 is the maximum and -3.8480 is the minimum

Answer 30

for which value of x did you get -3.8480?

Answer 31

-1.3660

Answer 32

I said that you have to consider the critical values that are present in the given limit , not outside values!!!!!!!!!!

Answer 33

ok

Answer 34

-1.3660 is out of limit [-1,1] right?

Answer 35

yea

Answer 36

So, now tell me what are the maxima and minima that you got in the given interval

Answer 37

while substituting the values of x for extrema, you have to substitute the upper limit and lower limit and also critical points in the interval

Answer 38

ok

Answer 39

tell me the values of extrema!

Answer 40

f(2) = 9 and f(-3) = -113 right?

Answer 41

f(-3) =49

Answer 42

ok

Answer 43

for the second problem . they are the extremas

Answer 44

ok

Answer 45

so the minimum is (-1.3660,-3.8480) and the max is (-3,49) ?

Answer 46

yes!

Answer 47

okay thanks

Answer 48

Your Welcome!