Find the minimum value of the quadratic below

Algebra Level pending

( x − 1 ) 2 + ( x − 2 ) 2 + ( x − 3 ) 2 + ( x − 4 ) 2 + ( x − 5 ) 2 + ( x − 6 ) 2 + ( x − 7 ) 2 (x - 1)^{2} + (x - 2)^{2} + (x - 3)^{2} + (x - 4)^{2} + (x - 5)^{2} + (x - 6)^{2} + (x - 7)^{2}

For real number x x , find the minimum possible value of the expression above.


The answer is 28.

Vijay Simha by Vijay Simha , 47, USA

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2 solutions

Vijay Simha
Mar 18, 2017

We know that this expression has to be a concave-up parabola (i.e. a parabola that faces upwards), and there is symmetry across the line x = 4. Hence, we can say that the vertex of the parabola occurs at x = 4. Plugging in this value for x in the equation above, we get 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28

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Zee Ell
Mar 18, 2017

Let y = x - 4

Then our expression becomes:

(y - 3)^2 + (y-2)^2 + (y - 1)^2 + y^2 + (y + 1)^2 + (y + 2)^2 + (y + 3)^2

After expanding and simplifying, we get:

5 y 2 + 28 → min ⁡ 5y^2 + 28 \rightarrow \min

Now it is easy to see, that this expression is minimal when y = 0 ( x = 4)

The minimum value: 28 \ \text {The minimum value: } \boxed {28}

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