Problem: Let y=(x−a)2+(x−b)2,a,b constants. For what value of x is y a minimum?
Answer Choices:
A. 2a+b
B. a+b
C. ab
D. 2a2+b2
E. 2aba+b
Solution:
y=(x−a)2+(x−b)2=2[x2−(a+b)x]+a2+b2
∴y=2[x2−(a+b)x+(2a+b)2]+a2+b2−2(2a+b)2
∴y=2[x−2a+b]2+2(a−b)2
For y to be a minimum x=\dfrac{a+b}
or
The graph of y=2x2−2(a+b)x+a2+b2 is a parabola concave up with respect to the x-axis. It has a minimum point on the axis of symmetry, the equation of which is x=−4−2(a+b), that is, x=\dfrac{a+b}