For all real numbers x find the minimum value of
(x + 12)^2 + (x + 7)^2 + (x + 3)^2 + (x - 14)^2 + (x - 8)^2
+506 differentiating with respect to x (using chain rule) gets:
2(x+12)+2(x+7)+2(x+3)+2(x−14)+2(x−8)=2(5x)=10x
There is only 1 critical point x=0, and since this function is concave up (the double derivative is positive), it is a minimum point. Plug in zero to your equation to get your answer.
+506 differentiating with respect to x (using chain rule) gets:
2(x+12)+2(x+7)+2(x+3)+2(x−14)+2(x−8)=2(5x)=10x
There is only 1 critical point x=0, and since this function is concave up (the double derivative is positive), it is a minimum point. Plug in zero to your equation to get your answer.
+2668 Here's another way, without using calculus:
Expand it to 5x2+462
Now, the minimum value occurs when x=0, because x2 is always non-negative.
