Finding the minimum in a given direction

Question

What's the manual way of finding function minimum in a given direction? The function is: $\min f(x)=x_1^2+x_2^2+x_1x_2 -x_2-x_1$ with the starting point in $x^0=[1,0]^T$ in a direction $d^0=[2,1]^T$.

Answer 1

Just replace $x_1 = 1 + 2t$ and $x_2 = 0 + 1t$. The resulting function on the line $x^0 + t d^0$ results in $f(t)= (1 + 2t)^2+ t^2+t(1+2t)-t-(1-2t)$. The first derivative results in $df/dt = 4(1+2t)+2t+1+4t-1-2 = 14t+2$. Consequently, you will find the minimum at $t=-2/7$ as the second derivative is greater than 0.