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\[f'(x) = 9x^2 -36\]
Setting that equal to 0 I get x=2 and x=-2

How do I know which is a max. and which is a min. ? Do I plug it into the original function?

|dw:1434785484976:dw| this is just an example...

still need help ?

you have to pick a point besides -2 and 2 in each interval

Ok so
f'(-3) = 45 (increasing)
f'(1) = -27 (decreasing)
f'(3) = 45 (increasing)
Is that right?

The calculator's giving me -27, is that wrong?

Let \(f(x)=3x^3−36x+3\)
-Find the points at which f achieves a local min. and local max.
find the derivative
\( \large f′(x)=9x^2−36\)
derivative = gradient... so when gradient = 0, its a turning poing
\( \large f′(0)=9x^2−36\)
\( \large f′(0)=2~ or~ -2\)
so what are the y values at these points?
\(f(x)=3x^3−36x+3\)
\(f(2)=3x^3−36x+3\)
\(f(2)=-45\) = local minima
\(f(-2)=51\) = local maxima

sorry.. yes that's right.. -27 for f(1) I hit the wrong number by accident

|dw:1434793225103:dw|

now we need the second derivative for concavity

\[\large f′(x)=9x^2−36\]
take the derivative of this equation again

@UsukiDoll Great! So the 2nd derivative is 18x right?

yes... so wee need to find the critical point
f''(x) = 18x
set the equation to 0 and solve for x

x=0 ?

Haha no worries... so can we choose x=-1 and x=1 ?

And I have to plug them in the 2nd derivative am I right?

yeah

f''(-1) = -18 and f''(1) = 18

Ohhh okie dokie! That makes a lot of sense now! :D ahaha

By this point have we found the points of inflection, or is there more to calculate?

-Find intervals on which f is concave up and concave down.
find the points either side of the local minima/maxima to work out if they're higher or lower
of either side is lower,
if the turning point is the highest point (highest y value), therefore it's concave up
if the turning point is the lowest point (lowest y value), its concave down
f(x)=3x^3−36x+3.
1st turning point = (-2, 51)
so a point either side of -2 is -1.9 and -2.1
y value at x = -1.9
\(\large f(x)=3x^3−36x+3\)
\(\large f(-1.9)=3x^3−36x+3\)
\(\large f(-1.9)=50.823\)
y value at x = -2.1
\(\large f(x)=3x^3−36x+3\)
\(\large f(-2.1)=3x^3−36x+3\)
\(\large f(-2.1)=50.817\)
each point either side is slightly lower than 51, so it's concave down
do the same for each side of the other turning point (2,-45) to work out what it is

I don't think so ... but I think |dw:1434793917549:dw|

Hahaha idk, I guess the 1st way you learn things always sits in the back of your mind better :D lol

hahahaa yeah... I still don't agree with his method XD

LOL XD

I agree, it's quite hard to agree with profs sometimes :P haha

-Find all points of inflection.
Let \(\large f(x)=3x^3−36x+3\)
find 2nd derivative
\(\large f(x)=3x^3−36x+3\)
\(\large f'(x)=9x^2−36\)
\(\large f''(x)=18x\)
when f''(x) = 0, it's a point of inflection
\(\large f''(x) = 18x\)
\(\large f''(0) = 18x\)
\(\large f''(0) = 0\) ==> so x = 0 is a point of inflection
what is the y value here?
\(\large f(x)=3x^3−36x+3\)
\(\large f(0)=3x^3−36x+3\)
\(\large f(0)=3\)
so point of inflection is at ( 0, 3)

Hahahaha right on!! XD above average grade!! I hope the same happens with me! :D

have you seen the questions from this one guy in the Qualified Helpers section? It's crazy! :/

we have the interval increasing or decreasing already.. via the chart.

And @UsukiDoll thank to you too!! I got everything you said! Including the 'chart' method! :D haha

And thanks for the medal! :D @UsukiDoll

xD thanks... everyone got a medal on here

Yeah, logic is always a saviour when it comes to exams >.< lol thanks again!!

np =)

:D

@UsukiDoll mind me asking which school you go to?

university

yeah lol same thing here in Canada... Which year are you? I'm a 1st year xD

I'm known as a super senior.. still going to school despite reaching over 120 credits

Well its gonna be my 2nd year this fall :D

Oh wow!!! 120 credits!!! Thats a lot!!!

I actually have 128...

Lucky you!!! :O

I need a MA in Math to teach in a college campus ...otherwise I would be stuck with high schoolers.

Hahahaha teaching high schoolers is a serious challenge XD lol

Specially when it comes to Math lol

oh yeah my state is the worst.

And wow! 6 years of college! That should take a lot hard work!!

lot of*