Question

Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 6x(1/3) + 3x(4/3). You must justify your answer using an analysis of f ′(x) and f ′′(x).

Answer 1

have you found f' and f'' yet?

Answer 2

also do you mean f(x)=6x^(1/3)+3x^(4/3)?

Answer 3

if you have found f' and f'' please share them

Answer 4

Taking calculus, as you apparently are, you should develop skills in presenting problems correctly. Freckles, above, has tried to find out whether you mean exponentiation.

One way of presenting math'l equations properly is through Equation Editor.

Your
\[f(x)=6x^(1/3)+3x^(4/3)\]

Answer 5

becomes \[ f(x)=6x ^{1/3}+3x ^{4/3}\]

Answer 6

Please differentiate this to obtain the first and second derivatives of f(x).

Answer 7

ok an dyes thats what i meant

Answer 8

power rule is your friend here
You will be using (x^n)'=n*x^(n-1) like crazy here

Answer 9

f'x = (4x+2)/x^2/3

Answer 10

that looks good for f'

Answer 11

f"(x) = (4x-4)/3x^5/3

Answer 12

that looks great too

Answer 13

the domain for our given function is all real numbers

now to find critical numbers you need to find when f'=0 and when f' does not exist

Answer 14

we will need to do this to find the relative extrema

Answer 15

so when f'=0 and when f' does not exist
we have the following two equations to solve:
\[4x+2=0 \\ x^{\frac{2}{3}}=0\]
You need to solve both of these equations (we will do the same thing for f'' to find the possible inflection points )

Answer 16

th first one is x=2/4

Answer 17

well 4x+2=0
4x=-2
x=-2/4
or reducing gives
x=-1/2
---

last equation should be easy x^(2/3) is 0 when x is 0


draw number line:


I prefer to use the first derivative to see when the function is increasing and decreasing to determine the max and min
we will test the 3 intervals above by choosing a number representative for each interval