Let $ d_{k} $ be a descent direction for the function $f$ in $ x_{k}\in\mathbb{R}^{n}$. Armijo criterion is $$f(x_{k}+td_{k})\leq f(x_{k})+\alpha t\nabla f(x_{k})^{T}d_{k}$$
I have to prove that if $\alpha>1$ then $\exists\,\varepsilon>0$ such that Armijo criterion is never satisfied when $0<t<\varepsilon$. Can anyone help me to prove that please? Thanks.
Hint: for $\alpha=1$ you have $y=f(x)+t\nabla f(x)^Td$ to be the tangent line to the graph of $\phi(t)=f(x+td)$ at $t=0$. When $\alpha>1$ then the line $y=f(x)+\alpha t\nabla f(x)^Td$ is lower than the tangent (note that $\nabla f(x)^Td<0$ as the direction is descent), so the graph to $\phi(t)$ for all small enough positive $t$ is going to be above the line.