When is the function f(x) = 3x^2 - 18x -100 at the lowest value
The extreme values of any function f(x) lie at the solution of the equation f'(x) = 0. If the solution is x = a and f''(a) is positive, this is a minimum value and if f''(a) is negative this is a maximum value.
For f(x) = 3x^2 - 18x - 100, f'(x) = 6x - 18
6x - 18 = 0
=> x = 3
f''(x) = 6 which is always positive.
The lowest value of f(x) lies at x = 3.