Question

Help Please!!!! (:
For x in [-13, 15] the function f is defined by
f(x)=x^6(x-6)^7

1.) On which two intervals is the function increasing.

2.) Find the region in which the function is positive.

3.) Where does the function achieve it's minimum?

Answer 1

first off, take derivative of f(x) what do you get?
second, consider the sign of the f'(x)
third, find the critical points, plug them back to f(x) , which smallest is minimum

Answer 2

\[f \prime (x)= 7(x-6)^6*(x^6)+6(x-6)^7*(x^5)\]

Answer 3

I solved for x and got
x=0 x=36/16 and x=6

Answer 4

ok, then ? consider the sign of f'(x)

Answer 5

what about the sign f'(x)?

Answer 6

hey, x =0, x =6 and x = 36/13, not 36/16

Answer 7

wait few second, I don't know whether it makes sense to you or not, I just post. I will scan it for you

Answer 8

I see where I went wrong with the 36/16 it's 36/13. haha and ok. (:

Answer 9

just to butt in for a second, it might be easier to analyze the sign of the derivative if you write it in factored form as \( x^5(x-6)^6 (13 x-36)\)

Answer 10

Wow! okay that was an easy way to do the problem by making that table! Thank you so much! I really appreciate the help!

Answer 11

that means you understand my way? ahahaaa... good

Answer 12

yes. sorta took me a while to understand the table. haha.

Answer 13

Could you help me with another similar problem. I understand the process. but the derivative has no solutions.

Answer 14

don't promise, just post

Answer 15

if I can, i will