calculus - Getting conflicting answers with the first derivative test...

I'm working with critical points and the 1st derivative test and am having a bit of confusion.

The original function

$$f(x) = x(2x-3)^{1/4}$$

has a derivative of

$$f'(x) = \frac{5x-6}{2(2x-3)^{(3/4)}}$$

Setting the numerator to 0 gives me a critical point of 6/5, while a 3/2 in the denominator makes it undefined. But, the graph can't exist at 6/5 = 1.2 because x must be greater than 1.5.

The red line is the original function, the purple curve is the derivative:

Furthermore, the derivative never equals 0, so it seems like there can be no critical points except for where f'(x) is undefined, namely, at 3/2. So there is only a single C.P. here.

When I go to make a line graph, all values to the left of 3/2 come up undefined, but all those to the right show than f(x) is increasing. So how can 6/5 be a critical point of the numerator in this case?

Doesn't this mean that there is no place where f(x) is decreasing? Perhaps there is no conflict here but I'm still not sure why the results are so weird. All the examples and practice problems using the line graph have always had either positive or negative values next to the critical point, instead of being undefined to one side.

Line graph with test values greater and less than 3/2:

DNE DNE  ...  ++++++++++++

<---|----------|--------|---->

    0         3/2       2