Question

find all intervals for which the graph of the function y=8x^3-2x^4 is concave downward. find all intervals for which the graph of the function y=8x^3-2x^4 is concave downward. @Mathematics

Answer 1

Use the second derivatie test

Answer 2

f'(x)=24x^2-8x^3
f''(x)=48x-24x^2

Answer 3

alright, that sounds familiar but then what?

Answer 4

find your inflection points by f''(x)=0

Answer 5

f''(x) =0 ?

Answer 6

okay got it, thanks.

Answer 7

so the function is convex for x > x- inflection point

Answer 8

wait what?

Answer 9

yes your inflection points are when f''(x)=0

Answer 10

i got inflection points x=0 and x=2

Answer 11

alright so just like testing for max and mins take a point from each side of your inflection points if
f''(c)>0 , it's concave upward
f''(c)<0 , concave downwards

Answer 12

so in this case i'd pick c=-1,1,3

Answer 13

yeah that's what i did. I got x<0 and x>2. thanks for the help.

Answer 14

so if you're looking for local minima and maxima, do you use the first or second derivative test?

Answer 15

first

Answer 16

to find the min/max take the crit pt of first derivative and put it in the original function

Answer 17

to get the inflection point take the crit pt of the 2nd derivative and put it in the original eq

Answer 18

okay thanks

Answer 19

lia i forgot this will u plz explain it to me

Answer 20

no problem

Answer 21

explain the question or the solution?

Answer 22

means solution

Answer 23

after finding the first derivative, make f'(x)=0. after finding the solutions to that, use points in between the points that you found to see where the graph of f'(x) will be negative or positive. wherever it's negative, it is concave downward.

Answer 24

thanks

Answer 25

f''(x)=0 is for inflection points or concavity.. Max and mins are for f'(x)

Answer 26

you can also use the 2nd dericative test dont forget. Where you take the crit points of f'(x) and plug them in to f''(x) which is easier way to test for relative extrema

Answer 27

There are all kinds of test.... so yeah pick one and enjoy lol