The function f(x) = ∑(x-r)^2, r ∈ (r = 1, 5) assumes minimum value at x = A. 5 B. 5/2 C. 3 D. 2 ​

Question

The function f(x) = \(\displaystyle\sum_{r=1}^{5} (x-r)^2\) assumes minimum value at x = 

A. 5 

B. \(\frac{5}{2}\)

C. 3 

D. 2

Answer 1

Option : (C)

\(\displaystyle\sum_{r=1}^{5} (x-r)^2\) 

f(x) = (x - 1)² + (x - 2)² + (x - 3)² + (x - 4)² + (x - 5)²

f’(x) = 2 [5x - 15] 

f’(x) = 0 ; x = 3 

Hence by second derivative test 

f’’(x) > 0 so it’s a point of minimum. 

f”(x) = 1 > 0 so At x = 3 minimum value.