By taking the derivative of the function. At the maximum or minimum of a function, the derivative is zero, or doesn't exist. And end-point of the domain where the function is defined may also be a maximum or minimum.
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Assuming the standard x and y axes, the range is the maximum value of y minus minimum value of y; and the domain is the maximum value of x minus minimum value of x.
vertex
A parabola's maximum or minimum is its vertex.
Standard notation for a quadratic function: y= ax2 + bx + c which forms a parabola, a is positive , minimum value (parabola opens upwards on an x-y graph) a is negative, maximum value (parabola opens downward) See related link.
Surely, you should check the value of the function at the boundaries of the region first. Rest depends on what the function is.