Prove that $f(x+y) \ge f(x+ty)$, if $f(x+y)$ is concave function and $t>1$.

Question

Let $f(x)$ be a concave function of many variables that we want to maximize $x=(x_1,x_2,\cdots,x_n)$ a point in its domain. Prove that if for some vector $y=(y_1,y_2,\cdots,y_n)$ we have that $f(x)\ge f(x+y)$ then for every number $t>1$, it holds that $f(x+y)\ge f(x+ty)$.

I know that $$f(\lambda x + (1 - \lambda)y)\ge \lambda f(x) + (1 - \lambda)f(y)\qquad \lambda\in[0,1]$$

I am having a really hard time, please help me. Thanks in advance.

Answer 1

Fix $t>1$.

Then in the inequality coming from the definition of $f$ being concave, replace $y$ with $x+ty$ and set $\lambda=\frac{t-1}{t}$