Question

The function \[ f(x)=\sum_{k=1}^{7}(x-k)^2 \] has a minimum value at \(x=a\). Then, \(a\) is equal to:

• A. 2
• B. 3/2
• C. 4
• D. 3/4

Hint:

For any function \( f(x) = \sum (x - k_i)^2 \), the value of \( x \) that minimizes the function is always the mean of the constants \( k_i \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find the value of \( x \) that minimizes the function \( f(x) = \sum_{k=1}^{n} (x - k)^2 \), we find the derivative \( f'(x) \) and set it to zero.

Step 2: Detailed Explanation:
\( f(x) = (x-1)^2 + (x-2)^2 + ... + (x-7)^2 \).
\( f'(x) = 2(x-1) + 2(x-2) + ... + 2(x-7) \).
Set \( f'(x) = 0 \):
\( 2 \sum_{k=1}^{7} (x - k) = 0 \implies \sum_{k=1}^{7} x - \sum_{k=1}^{7} k = 0 \).
\( 7x - \frac{7(7+1)}{2} = 0 \implies 7x - 28 = 0 \implies x = 4 \).

Step 3: Final Answer:
The minimum value occurs at \( a = 4 \).