How to find interval that's increasing/decreasing, local max/min values, interval of concavity, and inflection

The function f(x) = x^2 lnx

I need detailed steps on how to find:
1) intervals where f(x) is increasing/decreasing
2) local max/min
3)intervals of concavity
4)inflection points

Thank you very much!

f(x) = (x^2)ln(x)
f'(x) = 2x•ln(x) + x
f''(x) = 2ln(x) + 3

1.
2x•ln(x) + x = 0
x(2ln(x) + 1) = 0
=> x = 0 [but f is undefined for x ≤ 0, so this is not a critical point]
=> 2ln(x) + 1 = 0
2ln(x) = -1
ln(x) = -1/2
e^ln(x) = e^(-1/2)
x = 1/√e
f'(1/2) = ln(1/2) + 1/2 = -ln(2) + 1/2 < 0
f'(1) = 1 > 0
So f is increasing on (1/√e,∞) and decreasing on (0,1/√e).

2.
Given the results of (1), f has a local minimum at x = 1/√e and it has no local maximum.
f(1/√e) = (1/e)ln(1/√e) = (1/e)(-1/2)ln(e) = -1/(2e).
So the local minimum is at (1/√e,-1/(2e)).

3.
2ln(x) + 3 = 0
2ln(x) = -3
ln(x) = -3/2
e^ln(x) = e^(-3/2)
x = e^(-3/2).
f''(1/5) = 2ln(1/5) + 3 < 0
f''(1) = 3 > 0
So f is concave up on (e^(-3/2),∞) and concave down on (0,e^(-3/2)).

4.
Given (3), an inflection point occurs at x = e^(-3/2).
f(e^(-3/2)) = (e^(-3))ln(e^(-3/2)) = (e^(-3))(-3/2)ln(e) = -3/(2e^3).
So the inflection point is (e^(-3/2),-3/(2e^3)).