`f(x) = (x^2)ln(x)` (a) Find the intervals on which `f` is increasing or decreasing. (b) Find the local maximum and minimum values of `f`. (c) Find the intervals of concavity and the inflection...

`f(x) = (x^2)ln(x)` (a) Find the intervals on which `f` is increasing or decreasing. (b) Find the local maximum and minimum values of `f`. (c) Find the intervals of concavity and the inflection points.

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`f(x) = (x^2)ln(x)`

(a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points

`f(x)=(x^2)ln(x)`

(a) Take the derivative of the function.

`f'(x) =x^2 * 1/x + 2x * ln(x)`

`f'(x)= x +2xln(x)`

Set the derivative equal to zero.

`0=x+2xln(x)`

Factor the right side.

`0=x(1 + 2ln(x))`

Then, set each factor to zero and isolate the x.

For the first factor:

`x = 0`

For the second factor:

`1 + 2lnx = 0`

`2lnx=-1`

`lnx = -1/2`

`x =e^(-1/2)`
Hence, the critical numbers are `x=0` and `x=e^(-1/2) ` .

To get the intervals formed by these two critical numbers, we have to consider the domain of the function f(x). Take note that in logarithm, zero and negative numbers are not allowed. So the domain of f(x) is `(0, oo)` .

Considering the domain of the function, the intervals formed by the critical number are `(0, e^(-1/2))` and `(e^(-1/2),oo)` .

Then, assign a test value for each interval. Plug-in them to the...

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