Question

6x^3−27x^2−504x+7.

increasing and decreasing on what interval?

concave up and down and what interval?

Answer 1

first of all take first derivative

Answer 2

yea

Answer 3

what do I do with the first derivative

Answer 4

equate it to zero, and solve for x

Answer 5

ive got -4, and 7

Answer 6

okay we have ... 3 intervals

-inf, -4
-4,7
7, inf

what is the value of dy/dx in these three intervals??

Answer 7

i mean positive or negative.

Answer 8

do i just pick any number between -inf , -4 and plug into the original equation and see if it turns out positive or negative

Answer 9

in the dy/dx

Answer 10

ok 1 sec

Answer 11

i get a positive number for (7,inf) and a negative number for the other two

Answer 12

well, curve is increasing there.

Answer 13

ok but my software isnt accepting the ans...is this correct interval notation

(7,inf)

(-inf,4) U (-4,7)

Answer 14

choose any point in that interval and put it there, it will give you the characteristic of whole interval.

Answer 15

if we take 9 and plug into the derivative of 6x^3−27x^2−504x+7

it will give us the characteristics right, of (7,inf)

I got a positive number but my software isnt accepting (7,inf) as the increasing interval

Answer 16

yes,

Answer 17

Plug in a value into f'(x) for each interval if the out put is
f'(x) > 0 function is increasing
if
f'(x) < 0 function is decreasing.

For f''(x) do the same thing if the out put is
f''(x) > 0 the function is concave up
f''(x) < 0 the function is concave down

Answer 18

You find the intervals of increasing and decreasing by:

1. setting f'(x) = 0 and solving for x and using the x values you acquire to determine the the interval

2. Checking where the domain of f'(x) is undefinied. If f'(x) domain is undefinied in the same area as f(x). It is not a critical number and you shouldn't use it as an interval. If this is not the case it can be used as a critical number. and f''(x) = 0

You find the intervals of concavity by:

1. setting f''(x) = 0 and solving for x and using the x values you acquire to determine the the interval

2. Checking where the domain of f''(x) is undefinied. If f''(x) domain is undefinied in the same area as f(x). It is not a critical number and you shouldn't use it as an interval. If this is not the case it can be used as a critical number.

Answer 19

when I say use it as an interval I mean to define an interval

Answer 20

\[y'=18x^2-54x-504=18(x^2-3x-28)\]
(x-7)(x+4)=0
x=7, -4

Increasing: \[(-\infty,-4)U(7,\infty)\]