Question

find the point on the line y=x nearest to the point (4,1). THe ans. is (5/2,5/2)

Answer 1

geometry or calculus?

Answer 2

calculus

Answer 3

whew, then it is easy

Answer 4

ok show to me :D

Answer 5

any point on the line \(y=x\) will look like \((x,x)\) and the square of the distance between any point \((x,x)\) and \((4,1)\) is
\[f(x)=(x-4)^2+(x-1)^2\]

Answer 6

multiply out, combine like terms, take the derivative, set it equal to zero and solve for \(x\)

Answer 7

in fact you don't need calculus because \(f\) is a quadratic, and it will have a minimum at the vertex

Answer 8

you good from there?

Answer 9

ok :D thats all master ???

Answer 10

that is it, unless you need help with the algebra