Q:

Monotonicity of some function

I have a function $f: \mathbb R^n \rightarrow \mathbb R$ and nonzero $y\in \mathbb R^n$ such that
$$
f(z+ty) \leq f(z) \textrm{ for all } t\geq 0, z\in \mathbb R^{n}.  
$$
Is it then function $t \mapsto f(x+ty)$  nonincreasing for each fixed $x\in \mathbb R^{n}$ ?

A:

Fix $x$ and pick $t_1 <t_2$. Then setting $z= x+ t_1 y$ and $t = t_2 - t_1$, 
$$f( x+ t_1 y+ (t_2 - t_1)y) \leq f(x+ t_1 y) \Leftrightarrow f(x+ t_2 y)\leq f(x+ t_1 y)$$ 
So $t \mapsto f(x+ ty)$ is nonincreasing in $t$. 

