Final answer:
The maximum of the function f(x) = -2x² - 20x - 48 is found by using the vertex formula, which gives a vertex at x=5. Substituting this back into the function, the maximum value at this vertex is -198, which does not match any of the provided options.
The correct answer is none of all.
Step-by-step explanation:
The maximum of the quadratic function f(x) = -2x² - 20x - 48 can be found using the vertex formula since the coefficient of the x² term is negative, indicating that the parabola opens downwards and thus has a maximum point at its vertex. The x-coordinate of the vertex is given by the formula -b/(2a), where a is the coefficient of the x² term and b is the coefficient of the x term. In our case, a = -2 and b = -20.
Substituting these values into the formula gives us the x-coordinate of the vertex: x = -(-20)/(2 * -2) = -20/-4 = 5. To find the corresponding y-coordinate, which is the maximum value, we substitute x back into the function: f(5) = -2(5)² - 20(5) - 48 = -2(25) - 100 - 48 = -50 - 100 - 48 = -198.
However, there seems to be no possible correct answer in the options provided, since -198 is not listed. It's likely there was a mistake in either the function given or in the answer choices. The correct answer would be 'None of the above' if it were an option.