Question

Consider the function: f(x) = -(2/3)(X^3) + (9/2)(x^2) + 5x + 1 and find...

a) Inflection points (if any)

b) Critical values, if any

c) Identify the concavity, such as concave up or down, if any, and explain why.

d. show the rough graph to identify the extremes, such as min or max and inflection points.

Help me, please!!!!!!1

Answer 1

first lets multiply everything out

Answer 2

\[f(x)=\frac{-2}{3}x^3+\frac{9}{2}x^2+5x+1\]

Answer 3

next take the derivatie of this ... do you know how to?

Answer 4

\[\frac{d}{dx}[x^n]=nx^{n-1}\]

Answer 5

sorry, my machine is so slow. it doesnt let me reply fast. let me check what u wrote there

Answer 6

yes, -2x^2 + 9x + 5

Answer 7

what else do i do?

Answer 8

Pleaseeeeeeee, helpp!!

Answer 9

now solve \(y'=0\)

Answer 10

the solutions u find will be the critical points

Answer 11

what's the difference between critical points, critical values and critical numbers though?

Answer 12

x = 5, -1/2

Answer 13

well i guess they all mean the same.
the usual term is "critical point"

Answer 14

critical values i think mean the max and mins

Answer 15

u got it right!

Answer 16

i thought x was the critical number, y the critical value,and [x,y] the critical point

Answer 17

critical point (x,y)
critical number (x)
critical value (y)

Answer 18

you're correct

Answer 19

yes

Answer 20

yeah, i though so, lol. but i was confused as to how to get them. Should i work with the derivative or the original function/

Answer 21

i am confused*

Answer 22

critical number is the x at which y is a max or min value

Answer 23

to get a critical value you plug it into the original equation

Answer 24

we just got the critical numbers, right, and we used the derivative

Answer 25

yep now to get critical values plug the critical number into f(x)

Answer 26

so, for critical numbers we use f'(x) and for critical values f(x)?

Answer 27

yep

Answer 28

because f'(x) is the slope of the line

Answer 29

if you put your critical numbers into f'(x)= you get 0... that which is backwards of what you did to find the critical numbers

Answer 30

take x=5 for example if you put into f'(x) you get

\[-2(5)^2+5*9+5=-50+45+5=0\] which tells you that y' = 0

y' is \[\frac{dy}{dx}\] which is the change in y in respects to x which is known as slope

Answer 31

do you know how to get inflection points?

Answer 32

nope. that's what i really need help with

Answer 33

inflection points occur when f''(x)=0

Answer 34

so you need to take the derivative of f'(x)

Answer 35

-4x+9