The function \( f(x)=\sum_{k=1}^{7}(x-k)^2 \) has minimum value at \( x=a \), then \( a \) is equal to:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4
At a minimum the derivative is zero:
\( \displaystyle f'(x)=2\sum_{k=1}^{7}(x-k) \)
\( \displaystyle =2(7x-28) \)
\( 2(7x-28)=0\;\Rightarrow\;x=4 \)
Since \( f''(x)=14\gt 0 \), this is a minimum, so a = 4 (the mean of 1, 2, …, 7).