Question

helpppp: find the critical points: s(t)= (t-1)^4 (t+5)^3

Answer 1

@mathcalculus , what have you tried?

Answer 2

yes, i used the product rule:
\[\frac{ d }{ dx } (t-1)^{4}* (t+5)^{3}= (t-1)^{4} * 3(t+5)^{2} + (t+5)^{3} * 4(t-1)^{3}\]

Answer 3

then from there, i'm stuck.

Answer 4

ok, so far so good. Now, factor out the greatest common factor of both terms. For example, they both have a factor of (t-1)^3, but that's not all they have in common. What else?

Answer 5

?

Answer 6

Don't they both have a factor of t+5 to a power?

Answer 7

oh yes, but are from opposite sides..

Answer 8

doesn't matter the order. AB+CA=A(B+C)

Answer 9

can you explain to me. i dont really see where we are going with factoring

Answer 10

The purpose in factoring is to simplify the derivative into a product of three factors. One of the factors will be t-1 to a power, a second factor will be t+5 to a power, and then you will be left with some factor containing a t as well.

Answer 11

And then once you have your derivative in factored form, you can set each factor equal to zero to find your critical points.

Answer 12

can you show me

Answer 13

i understand what youre saying but not how sure how to do that.

Answer 14

Ok, lets look at the t-1 term in each expression.

Answer 15

ok

Answer 16

What is the largest exponent they share?

Answer 17

3

Answer 18

now i know that i write (t-1)^3

Answer 19

but what happens to the 4?

Answer 20

Exactly, so, we may factor out a \[\left( t-1 \right)^{3}\] from both expressions. Hang on, we will come to that.

Answer 21

k

Answer 22

Let's take a looke at the first expression in its entirety. If all you are removing is the \[\left( t-1 \right)^{3}\]

You would be left with \[4\left( t+5 \right)^{3}\]

Answer 23

Do you see that?

Answer 24

no if i was to see the common factor.. i would get (t-1)^3+ (t+5)^2.... then what do i do with the numbers that are left...?

Answer 25

ok,, there is your mistake. It isn't a sum. It's a product.

Answer 26

ok so i dont see how you got 4(t+5)^3

Answer 27

Hang on, I was only talking about factoring out the t-1 from the first expression,, not from the whole thing.

Answer 28

well you showed me that

Answer 29

Hang on my pc is giving me fits.

Answer 30

kk