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 minimum value of a function defined as a sum of squared deviations \( \sum (x - k)^2 \), we can use the property that the sum of squared deviations is minimized at the arithmetic mean of the values \( k \).

Step 2: Mathematical Calculation:
The function is \( f(x) = \sum_{k=1}^{7} (x - k)^2 \). Setting the derivative \( f'(x) = 0 \):
\( f'(x) = \frac{d}{dx} \sum_{k=1}^{7} (x^2 - 2kx + k^2) = \sum_{k=1}^{7} (2x - 2k) = 14x - 2 \sum_{k=1}^{7} k = 0 \).
\( 14x = 2 \times \frac{7(1+7)}{2} = 56 \).
\( x = \frac{56}{14} = 4 \).

Step 3: Final Answer:
The function reaches its minimum at \( a = 4 \).