Question

For supplemental problem 2, G-2 : http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-a-functions-of-two-variables-tangent-approximation-and-optimization/problem-set-4/MIT18_02SC_SupProb2.pdf ,

Isn't it enough to simply show the equation always has solutions as long as the determinant is not zero? And that if all the xi's are the same, the determinant will be zero, and thus no solution (in a vertical line)?

In other words, what does the Hint mean? How do we apply the Hint to solve the problem?

Answer 1

The "equation (4)" they refer to in the question is from
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/readings/least_sq_intrpol.pdf
which in turn is found at
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/readings/supp_notes/

Answer 2

Equation 4 will always have a solution as long as the deternimnant of \[\left[\begin{matrix}s ^{-} & x^{-} \\ x^{-} & 1\end{matrix}\right]\] is not equal to zero right?
So what does the hint: "Hint: use the fact that for all values of u we have \[\sum_{i=1}^{n}(x _{i}-u)^{2}\ge0\] since squares are always non-negative. Write the left side as a quadratic polynomial in u. Usually it has no roots. What does this imply about the coefficients? When does it have a root? (answer these two questions by using the quadratic formula)"

Isn't all that unnecessary?

Answer 3

*so what does the hint mean? Isn't all that unnecessary?

Answer 4

Ok so here's my reasoning-
Since for \[f(u) = \sum_{i=1}^{n}(x _{i}-u)^{2}= n u ^{2}+u(2\sum_{i=1}^{n}x _{i}) + \sum_{i=1}^{n}x _{i}^{2}\ge0\], f(u) will have at most one root when the discriminant is equal to zero. This means that the discriminant is always less than or equal to 0, i.e. \[(2\sum_{i=1}^{n}x _{i})^{2}-4n \sum_{i=1}^{n}x _{i}^{2}\le0\] i.e. usually, \[n \sum_{i=1}^{n}x _{i}^{2} -(\sum_{i=1}^{n}x _{i})^{2}>0\]This is the determinant for equation 4. The fact that this is usually >0 means that in most cases there will be a solution. The solution will only not exist when \[n \sum_{i=1}^{n}x _{i}^{2}= (\sum_{i=1}^{n}x _{i})^{2}\], i.e. when all xi are the same. i.e. it is a vertical line

Is this reasoning correct?

Answer 5

I assume by the "determinant of equation 4" you mean after you put the equations in matrix form?
But yes, your argument looks good.