I need help with this problem.
Prove the concavity of $F(x,y) = \ln(x) + y$ by arguing the definition of concavity.
A function $f$ is concave is for any $x_0, x_1 \in \mathbb{R}^2$ and $t \in [0,1]$, $$
f((1 – t) x_0 + t x_1) \geq (1 – t) f(x_0) + t f(x_1)
$$
Also, I am not allowed to use the theorem that the sum of concave functions is concave.
Best Answer
You need to show that for any $\langle x_0,y_0\rangle$ and $\langle x_1,y_1\rangle$ and any $t\in[0,1]$,
$$\ln\big(tx_0+(1-t)x_1\big)+ty_0+(1-t)y_1\ge t(\ln x_0+y_0)+(1-t)(\ln x_1+y_1)\;;$$
after a little simplification this becomes
$$\ln\big(tx_0+(1-t)x_1\big)\ge t\ln x_0+(1-t)\ln x_1\;.$$
If you already know that the log is concave, you’re done at this point.