How to find the coordinates of a stationary point? PLEASE HELP?

Stationary points are where dy/dx = 0, makes sense in thinking about a maximum or minimum, in order for them to be a maximum or minimum, then their gradient must be zero.

In this example, you're going to have to differentiate using the product rule, as it's in the form:

y = uv

dy/dx = vu' + uv'

so y = 4x^2lnx where u = 4x^2 and v = ln x note that the differential of lnx is 1/x

and dy/dx = 8xlnx + (4x^2 x 1/x)

= 8xlnx + 4x

dy/dx = 0 so.... 8xlnx = -4x

2xlnx = -x

2lnx = -1

lnx = -1/2

x = e^-1/2 = 0.607

and so y = 4(0.607)^2 x ln0.607 = -0.736

To determine whether it's a maximum or minimum stationary point, consider the following:

If the second differential, that is, the changing gradient, is positive, then the line is on its way up, it was at a minimum point, whereas if the changing gradient is negative, then the line is on its way down, it was at a maximum point.

so we had dy/dx = 8xlnx + 4x

d^2y/dx^2 = 8lnx + 8 + 4

substitue in the value of x of 0.607

d^2y/dx^2 = 8ln(0.607) + 8 + 4 = 8

This is positive, therefore the stationary point is a minimum point.

Hope this was of help :)