PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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Solution for The Function F(X) = 5 ∑ R = 1 (X − R)2 Assumes Minimum Value at X = (A) 5 (B) 5 2 (C) 3 (D) 2 - PUC Karnataka Science Class 12 - Mathematics

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Question

The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x =

(a) 5 
(b) \[\frac{5}{2}\]

(c) 3
(d) 2

Solution

\[(c) 3\]

\[\text { Given:} f\left( x \right) = \sum^5_{r = 1} \left( x - r \right)^2 \]

\[ \Rightarrow f\left( x \right) = \left( x - 1 \right)^2 + \left( x - 2 \right)^2 + \left( x - 3 \right)^2 + \left( x - 4 \right)^2 + \left( x - 5 \right)^2 \]

\[ \Rightarrow f'\left( x \right) = 2\left( x - 1 + x - 2 + x - 3 + x - 4 + x - 5 \right)\]

\[ \Rightarrow f'\left( x \right) = 2\left( 5x - 15 \right)\]

\[\text { For a local maxima and a local minima, we must have }\]

\[f'\left( x \right) = 0\]

\[ \Rightarrow 2\left( 5x - 15 \right) = 0\]

\[ \Rightarrow 5x - 15 = 0\]

\[ \Rightarrow 5x = 15\]

\[ \Rightarrow x = 3\]

\[\text { Now,} \]

\[f''\left( x \right) = 10\]

\[f''\left( x \right) = 10 > 0\]

\[\text { So, x = 3 is a local minima }. \]

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Solution The Function F(X) = 5 ∑ R = 1 (X − R)2 Assumes Minimum Value at X = (A) 5 (B) 5 2 (C) 3 (D) 2 Concept: Graph of Maxima and Minima.
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