The given function is a quadratic function in the form of f(x) = ax^2 + bx + c. The vertex form of a quadratic function is given as g(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
In this case, 'a' is the coefficient of x^2, 'b' is the coefficient of x, and 'c' is the constant term. The vertex of the parabola is given by the point (h, k), where h = -b/(2a) and k = f(h).
Given the coefficients as a=2, b=24, and c=83, we first find the x-coordinate of the vertex(h):
h = -b/(2a) = -24/(2*2) = -24/4 = -6.
Substitute h = -6 into f(x) to find the y-coordinate (k):
k = f(h) = 2*(-6)^2 + 24*(-6) + 83 = 72 - 144 + 83 = 11.
So, the vertex is (-6, 11).
In a parabola, if the coefficient of x^2 ('a') is positive, the parabola opens upwards and hence the vertex point represents a minimum point because it is at the bottom of the parabola.
Given 'a' = 2, which is a positive integer, it means the parabola opens upwards. Therefore, the vertex of this function, (-6, 11), is a minimum point.
In summary, for the given function f(x) = 2x^2 + 24x + 83, the vertex is (-6, 11) and it represents a minimum point.