Explanation:
g(x) = x² - 2x + 8
the maximum is easy. as x² had a much bigger progression than 2x with increasing x, when x goes against + or - infinity, g(x) will go to +infinity.
the minimum is the point where the squared function curve turns around.
we get this the resist way by the first derivative of the function and finding its "zero" (the value of x for which the first derivative is 0).
g'(x) = 2x - 2
2x - 2 = 0
2x = 2
x = 1
so, the extreme point (in this case the minimum) is at x = 1.
and what is g(1) ?
g(1) = 1² -2×1 + 8 = 1 - 2 + 8 = 7
so, the minimum of g(x) is 7.