Final Answer:
The function f(x) = 8x² - 40 has a minimum value.
Step-by-step explanation:
To determine whether the function has a minimum or maximum value, let's examine its nature based on its coefficient of x². In the given function f(x) = 8x² - 40, the coefficient of x² is positive (8), indicating that the parabola opens upwards, which means the function has a minimum value.
The general form of a quadratic function is f(x) = ax² + bx + c, where 'a' is the coefficient of x². When 'a' is positive, the parabola opens upwards, resulting in a minimum value for the function. In this case, 'a' is 8, which is positive, confirming that the function has a minimum value.
The function f(x) = 8x² - 40 represents a parabola that opens upward, implying the existence of a minimum point. This minimum point occurs at the vertex of the parabola, which is the lowest point on the graph. To find the x-coordinate of the vertex, we use the formula x = -b / (2a) for the quadratic equation in the form f(x) = ax² + bx + c. Here, a = 8 and b = 0 (since there's no x-term). Plugging these values into the formula gives x = -0 / (2 * 8) = 0. Thus, the minimum value occurs at x = 0, and substituting x = 0 into the function gives f(0) = 8(0)² - 40 = -40, indicating that the minimum value of the function is -40.