Step 1: Understand the Concept:
To determine the minimum or maximum value of each function in List-I, analyze their properties. This requires understanding the range of basic functions such as squares, absolute values, and trigonometric functions.
Step 3: Detailed Explanation:
(A) f(x) = $(2x - 1)^2 + 3$:
The term $(2x - 1)^2$ is a squared expression, thus its minimum value is 0. This minimum is achieved when $2x - 1 = 0$.
Consequently, the minimum value of the function is $0 + 3 = 3$. This corresponds to option (III).
(B) f(x) = $-|x + 1| + 4$: (Note: The OCR may have missed a preceding negative sign).
The term $|x + 1|$ is always non-negative, with a minimum value of 0.
Therefore, $-|x + 1|$ has a maximum value of 0.
The maximum value of the function is $0 + 4 = 4$. This corresponds to option (I).
(C) f(x) = sin(2x) + 6:
The sine function, sin(2x), has a range of [-1, 1].
The minimum value of sin(2x) is -1.
Therefore, the minimum value of the function is $-1 + 6 = 5$. This corresponds to option (IV).
(D) f(x) = $-(x - 1)^2 + 10$:
The term $(x - 1)^2$ has a minimum value of 0.
Consequently, $-(x - 1)^2$ has a maximum value of 0.
The maximum value of the function is $0 + 10 = 10$. This corresponds to option (II).
Step 4: Final Answer:
The correct pairings are (A) - (III), (B) - (I), (C) - (IV), and (D) - (II). This matches option (3).