Can anyone shed a light on this problem?
Let $$f(x) = x^t A x + b^t x + c$$ in which $A \in \mathbb R^{nxn}$ is symmetric and positive definite, $b \in \mathbb R^n$ and $x \in \mathbb R^n$. Let $L_1$ and $L_2$ be two parallel straight lines in $\mathbb R^n$ and $d$ be their direction vector. Let $x^1$ and $x^2$ be the minimizer of $f$ in $L_1$ and $L_2$, respectively. Prove that $$(x^2 - x^1)Ad = 0$$
This problem is in the section on exact line search, steepest descent, and Newton method.
Let $\phi(t) = f(x+td)$, then $\phi'(t) = 2x^TAd+2t d^TAd+ b^T d$.
In particular, $\phi'(t) = 2(x+td)^T Ad + b^T d$ and if $t$ is a minimiser and $y=x+td$, we have $2 y^T Ad + b^T d = 0$.