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

Question

tmagloire1 · Answer

@phi @Directrix

tmagloire1 · Answer

@amistre64

amistre64 · Answer

define f' and f''

tmagloire1 · Answer

f' = (1+8 x)/x^(2/3)
f'' = (2 (-1+4 x))/(3 x^(5/3))

amistre64 · Answer

f = 3x^(1/3) + 6x^(4/3)

f' = 3/3 x^(-2/3) + 6*4/3 x^(1/3)

f'' = 3/3* -2/3 x^(-5/3) + 6*4/3*1/3 x^(-2/3)

how do we analyse them?  what defines the critical points?

tmagloire1 · Answer

Critical points are when you set them = 0 and see where they change direction?

amistre64 · Answer

0 is only one of the steps ... also seeing where they are undefined is another

tmagloire1 · Answer

How do you find where they're undefined?

amistre64 · Answer

what makes a fraction undefined?

amistre64 · Answer

x^(-n) = 1/x^n

but this is not defined for what value of x?

tmagloire1 · Answer

anything divided by 0

amistre64 · Answer

then in this case that is going to be our undefineds

tmagloire1 · Answer

How does that relate to the problem though?

amistre64 · Answer

that is how you "Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x)"

f' and f'' = 0 or undefined ...

tmagloire1 · Answer

Oh ok so is there a different way to find extrema and inflection points?

amistre64 · Answer

precisely .. no

graphing utility maybe, but by eye .. no

tmagloire1 · Answer

so then extrema and inflection points are the same thing?

amistre64 · Answer

no

extrema are your highest/lowest ponts

inflection is where slope changes direction

amistre64 · Answer

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extrema, A and B

inflection, M