Thread: Find Where f Increase, Decrease, Local Minimum, Inflection Point, concave up and down

f(x) = x^3 lnx

f'(x) = x^2 (3lnx +1)

x= e^(-1/3)

left side of e^(-1/3) is negative, right side is positive so

f increase at (e^(-1/3), infinity)
f decrease (-infinity , e^(-1/3)) correct?

Local Minimum:
f(e^(-1/3)) = .123 so local minimum is:
.123 correct?


Inflection point:

f'(x) = x^2 (3lnx +1)

f "(X) = 6xlnx + 3x + 2x
= x (6lnx + 5) = 0

x= e^(-5/6)

----- e^(-5/6) ++++

I don't know what to do from here..

Originally Posted by NeoSonata

f increase at (e^(-1/3), infinity)

Right.

f decrease (-infinity , e^(-1/3)) correct?

Wrong, is defined in . As a consequence, it is decreasing in .

Originally Posted by NeoSonata

Inflection point:

f'(x) = x^2 (3lnx +1)

f "(X) = 6xlnx + 3x + 2x
= x (6lnx + 5) = 0

x= e^(-5/6)

----- e^(-5/6) ++++

I don't know what to do from here..

You have found wherefore . So what's the inflection point (x and y coordinate)? ...
You can now determine where the functions is convex or concave.

we have to check f '
f ' is negative if <
gives x <
therefore f is decreasing if x ( 0 , )
and f is increasing if x ( , )