Question

Which of the following functions have a maximum value?

A) S(x) = 2(x + 2)^2 - 2
B) 8(x) = -21x^2 - 48x - 4
C) n(x) = -1x^2 - 2x + 3
D) }(x) = -(x + 1)^2 - 2
E) R(x) = x^2 - 4
F) m(x) = 2(x + 1)^2 - 2

Select the correct options:
a) A and F
b) B and D
c) C and E
d) None of the above

Answer 1

The functions B, 8(x) = -21x^2 - 48x - 4, and C, n(x) = -1x^2 - 2x + 3, as well as D, }(x) = -(x + 1)^2 - 2, have a maximum value because their quadratic terms have negative coefficients, indicating downward-opening parabolas.

Step-by-step explanation:

To determine which functions have a maximum value, we need to analyze the coefficient of the quadratic term in each function. A maximum value occurs in a quadratic function when the parabola opens downward, which means the coefficient of the x^2 term should be negative.

• S(x) = 2(x + 2)^2 - 2: The coefficient of the quadratic term is positive (2), so it does not have a maximum.
• 8(x) = -21x^2 - 48x - 4: The coefficient of the quadratic term is negative (-21), so it has a maximum.
• n(x) = -1x^2 - 2x + 3: The coefficient of the quadratic term is negative (-1), so it has a maximum.
• } (x) = -(x + 1)^2 - 2: This function is in the form of a downward-opening parabola, indicating a maximum value.
• R(x) = x^2 - 4: The coefficient of the quadratic term is positive (1), so it does not have a maximum.
• m(x) = 2(x + 1)^2 - 2: The coefficient of the quadratic term is positive (2), so it does not have a maximum.

Therefore, the correct options that have a maximum value are:

1. 8(x) = -21x^2 - 48x - 4
2. n(x) = -1x^2 - 2x + 3
3. } (x) = -(x + 1)^2 - 2