This question requires us to match functions given in List-I with their corresponding minimum or maximum values from List-II. Let's analyze each function to find its correct match.

Analyzing Function Minimum Value: $f(x) = (2x-1)^2+3$

For the function $f(x) = (2x-1)^2+3$, we need to find its minimum value.

• The term $(2x-1)^2$ is a square of a real number. The square of any real number is always non-negative, meaning it is greater than or equal to zero.
• So, the minimum value of $(2x-1)^2$ is $0$.
• This minimum occurs when $2x-1 = 0$, which implies $x = \frac{1}{2}$.
• Therefore, the minimum value of the function $f(x)$ is $0 + 3 = 3$.
• This matches with List-II (III). So, (A) corresponds to (III).

Finding Function Maximum Value: $f(x) = -|x + 1| + 4$

For the function $f(x) = -|x + 1| + 4$, we need to find its maximum value.

• The term $|x + 1|$ represents the absolute value of $(x+1)$. The absolute value of any real number is always non-negative, meaning it is greater than or equal to zero.
• So, the minimum value of $|x + 1|$ is $0$. This occurs when $x+1=0$, which means $x=-1$.
• Consequently, the term $-|x + 1|$ will have a maximum value of $0$ (when $|x+1|=0$).
• Therefore, the maximum value of the function $f(x)$ is $0 + 4 = 4$.
• This matches with List-II (I). So, (B) corresponds to (I).

Determining Function Minimum Value: $f(x) = \sin(2x) + 6$

For the function $f(x) = \sin(2x) + 6$, we need to find its minimum value.

• The sine function, $\sin(\theta)$, has values that range between $-1$ and $1$, inclusive.
• The minimum value that $\sin(2x)$ can take is $-1$.
• Therefore, the minimum value of the function $f(x)$ is $-1 + 6 = 5$.
• This matches with List-II (IV). So, (C) corresponds to (IV).

Calculating Function Maximum Value: $f(x) = -(x-1)^2+10$

For the function $f(x) = -(x-1)^2+10$, we need to find its maximum value.

• The term $(x-1)^2$ is a square of a real number, so its minimum value is $0$. This occurs when $x-1 = 0$, which means $x=1$.
• The term $-(x-1)^2$ will therefore have a maximum value of $0$ (when $(x-1)^2=0$).
• Thus, the maximum value of the function $f(x)$ is $0 + 10 = 10$.
• This matches with List-II (II). So, (D) corresponds to (II).

Summary of Matches

Based on the analysis, the correct matches are:

• (A) The minimum value of $f(x) = (2x-1)^2+3$ is 3, matching (III).
• (B) The maximum value of $f(x) = -|x + 1| + 4$ is 4, matching (I).
• (C) The minimum value of $f(x) = \sin(2x) + 6$ is 5, matching (IV).
• (D) The maximum value of $f(x) = -(x-1)^2+10$ is 10, matching (II).

Therefore, the correct option is (A) - (III), (B) - (I), (C) - (IV), (D) - (II).