Triangle lower bound of a convex function

Question

Suppose the function $f:[0,\infty)\rightarrow [0,\infty)$ is convex on $[T,\infty)$.

Let $\delta=f(t)/|f'(t)|$ then the triangle $T$ determined by the points $(t,f(t)), (t,0)$ and $(t,t+\delta)$ lies below the graph of $f$ for all $t\geq T$.

---

I cannot confirm that the point $(t,t+\delta)$ does lie below the graph of $f$. I tried to play around the tangent definition of convex function: $$ f(x)\geq f(y)+f'(y)(x-y) $$ but unable to get anywhere.

---

(Edit: I have attached a picture of the source material 06-01-2021)

Answer 1

In case $f'(t)=0$, one can check either the integral is infinity, or the integral is $0$, meaning $f=0$ on $[t, \infty)$ and $f(t)=0$, you can interpret $f(t)^2/|f'(t)|=0$.