The problem description is as follows:
Suppose there is a concave function $y(x)$. Now, suppose that we're interested in plotting a curve $g(t) = \sup[y - tx]$. Prove that $g(dy/dx)=y_t-tx_t$, where $x_t$ and $y_t$ are the coordinates of the point on the curve where the slope $y'$ is equal to $dy/dx$.
This is easy to prove graphically, but I have been struggling to do it analytically.
Graphically, since $g$ is (on the original graph) a supremum of the y-intercepts of straight lines with a predefined slope $t$, and since the slope of the curve is non-increasing, drawing the straight line with slope $t$ as a tangent to the curve (i.e., at the point where the slope of the curve itself is equal to $t$) is optimal because as we move away from that point, the amount that $g$ decreases is more than the slope $t$ can recover because of concavity, causing a net drop in the resulting $g$ (and if it increases, it increases by a lower amount than the slope $t$ is able to do). Since we're looking for a maximal $g$, this means that the point on the curve with slope $t$ defines the optimal line we're looking for.
Analytically, I thought it would be easy to say the same thing I did graphically, but it hasn't worked. My approach was to assume that a point $(k_s,q_s)$ defines the optimum, where the slope of the curve at that point is not equal to $t$. Then, using the concavity definition, I would derive a contradiction and show that that means that the slope has to be $t$.
Here's the start of what I was thinking:
$y(\lambda x_s+(1-\lambda)x_t)\geq\lambda y(x_s)+(1-\lambda)y(x_t)$
Subtracting $(\lambda x_s+(1-\lambda)x_t)t$ from both sides:
$y(\lambda x_s+(1-\lambda)x_t)-(\lambda x_s+(1-\lambda)x_t)t\geq\lambda y(x_s)+(1-\lambda)y(x_t)-(\lambda x_s+(1-\lambda)x_t)t$
I proceeded to try and manipulate both sides of the inequality to get a contradiction. For example, I tried simplifying the right-hand side to terms of $k_t$ only, but it got me nowhere. If I could do the same but keep $k_s$ terms instead, that would have been a contradiction, but it wasn't working. I also tried taking the left-hand side in terms of any point $k'$ between $k_t$ and $k_s$, but that got me nowhere too.
I feel like the answer is so simple, but it's driving me crazy.
I did try some other approaches too, but I think they're useless...
By the way, I also know (based on my intuitive and graphical understanding of the problem - I've actually even already constructed a graphical proof) that the resulting graph of g is convex, and I'll be trying to prove it analytically afterwards. So, if you happen to prove that analytically too somehow, that would be great!