- #1
Bosley
- 10
- 0
Homework Statement
Suppose F: Rn --> R has first order partial derivatives and that x in Rn is a local minimizer of F, that is, there exists an r>0 such that
f(x+h) [tex]\geq[/tex] f(x) if dist(x, x+h) < r. Prove that
[tex]\nabla[/tex] f(x)=0.
Homework Equations
We want to show that fxi(x) =0 for i = 1,...,n
So we want to show that [tex]\lim_{t\to 0}\frac{f(x + t e_i) - f(x)}{t} = 0[/tex]
Where [tex]e_i[/tex] is the ith standard basis element.
The Attempt at a Solution
We know f(x+h) [tex]\geq[/tex] f(x) if ||(x+h) - x|| <r, that is, if ||h|| < r.
Consider |t| < r. Then ||t ei|| = |t| < r.
So then f(x) [tex]\leq[/tex] f(x + t ei) for all t such that |t| < r, and f(x+t ei) - f(x) [tex]\geq[/tex] 0.
But I don't know where to go from here...insight?