Question

Rolle's Theorem
f(x)=(x-2)(x-3)(x-4) on [2,4]

Answer 1

did you satisfy the hypotheses of rolle's theorem, first?

Answer 2

it must be differentiable on (2,4) and continuous on [2,4] , and it is since it is a polynomial

Answer 3

thats what I don't know how to do

Answer 4

Oh okay I wasn't really sure that it was a polynomial

Answer 5

its a polynomial in factored form (linear factors)

Answer 6

so then plug in 2 and 4

Answer 7

and then take the derivative to find c?

Answer 8

first plug in f(2) and f(4)

Answer 9

rolles theorem says we must have f(a) = f(b)

Answer 10

okay well I got -2 for f(2) and 0 for f(4) so rolle's theorem does not apply right?

Answer 11

should be zero for f(2)

Answer 12

if you plug in 2 , you have (2-2) * whatever = 0

Answer 13

oops i see what I did wrong

Answer 14

so then i just take the derivative and find c right?

Answer 15

so now since we have f(2) = f(4) , then there exists a value c inside (2,4) such that f ' (c) = 0

Answer 16

correct, take derivative

Answer 17

the derivative is just 1 right?

Answer 18

the derivative should not be 1

Answer 19

but the derivative for each is 1 isn't it?

Answer 20

yes but youre using product rule incorrectly

Answer 21

so then it's just (1)(1)(1)

Answer 22

what am I doing wrong?

Answer 23

well

Answer 24

you want to use product rule?

Answer 25

here is the product rule

( u * v ) ' = u' * v + u * v '

Answer 26

yes I think

Answer 27

we have three factors here though

Answer 28

ohhh I thought i couldve just taken the derivative of each

Answer 29

( u * v * w ) ' = u ' * v * w + u * v ' * w + u * v * w '

Answer 30

i dont think this is the easiest way to differentiate it , though

Answer 31

ohh there's an easier way?

Answer 32

yes we can expand the original f(x)

Answer 33

product rule?!

Answer 34

technically you could use the product rule :)

Answer 35

but will be more work

Answer 36

so what should I do?

Answer 37

f(x)=(x-2)(x-3)(x-4)
= ( x^2 -2x -3x + 6) (x-4)

Answer 38

so first expand (x-2)(x-3) using foil

Answer 39

oh and then use the product rule from there?

Answer 40

then multiply by x - 4

Answer 41

(x-2)(x-3)(x-4)
= ( x^2 -2x -3x + 6) (x-4)
= ( x^2 - 5x + 6) ( x - 4)
= x^3 - 5x^2 + 6x - 4x^2 +20x - 20

Answer 42

f(x) = x^3 -8x^2 + 26x - 20

Answer 43

ok? foil then foil again

Answer 44

and then I take the derivative?

Answer 45

yes

Answer 46

now its easy to take the derivative

Answer 47

ohh okay thank you so much