1. Sketch x^(1/3) * e^-x/3

I am having trouble finding a compact formula for y'', so that I can place it on my sign table, but instead i got a really long equation which is too big

2. Originally Posted by SyNtHeSiS
I am having trouble finding a compact formula for y'', so that I can place it on my sign table, but instead i got a really long equation which is too big
hi

Yeah , i tried the second derivative too and ended up with sth really long and uninteresting .

Fisrtly , find its first derivative which is

$y'(x)=\frac{e^{-\frac{1}{3}x}(x-1)}{3x^{\frac{2}{3}}}$

when y'(x)=0 , x=1

Make a table calculating dy/dx for 1 , 1^- , 1^+

this point (1 , $\frac{1}{e^{\frac{1}{3}}}$) is a local maximum .

And also the graph passes thought point (0,0)

Lets take a look at the behaviour of the graph as it goes to +/- infinity

when x approaches +ve infinity , y approaches 0

When x approaches -ve infinity , y approaches -ve infinity .

And it seems that (0,0) is a point of inflexion here .

3. For my first derivative I got y' = -1/3x ^ (1/3) * e^(-x/3) * (1 - 1 / x). I still dont know a simpler way to find y'' cause the normal way too long.