Answer: To determine whether the given functions have a minimum value or a maximum value, we can use the concept of concavity and the coefficient of the leading term.
The coefficient of the leading term (-4x^2) is negative, which means the parabola opens downwards. In this case, the function has a maximum value. To find the value, we need to locate the vertex of the parabola. The vertex of a parabola in the form f(x) = ax^2 + bx + c is given by the coordinates (h, k), where h = -b / 2a and k is the maximum (or minimum) value of the function.
For f(x) = -4x^2 - 24x + 15:
a = -4, b = -24
h = -(-24) / (2 * (-4)) = 24 / 8 = 3
Now, substitute h back into the function to find the maximum value (k):
f(3) = -4(3)^2 - 24(3) + 15
f(3) = -4(9) - 72 + 15
f(3) = -36 - 72 + 15
f(3) = -93
So, the function f(x) = -4x^2 - 24x + 15 has a maximum value of -93.
The coefficient of the leading term (6x^2) is positive, which means the parabola opens upwards. In this case, the function has a minimum value. Again, we can find the value by locating the vertex of the parabola.
For f(x) = 6x^2 + 36x - 20:
a = 6, b = 36
h = -36 / (2 * 6) = -36 / 12 = -3
Now, substitute h back into the function to find the minimum value:
f(-3) = 6(-3)^2 + 36(-3) - 20
f(-3) = 6(9) - 108 - 20
f(-3) = 54 - 108 - 20
f(-3) = -74
So, the function f(x) = 6x^2 + 36x - 20 has a minimum value of -74.