From Convex Optimization by Boyd & Vandenberghe:
Let $f: \Bbb R^n \rightarrow \Bbb R$ be continuous and twice differentiable.
Given this inequality
$$\tilde f''(t) \le \tilde f''(0) + tL \|\Delta x_{\text{nt}}\| \le \lambda(x)^2 + t\frac{L}{m^{3/2}}\lambda(x)^3 $$ where $$\tilde f''(0) = \lambda(x)^2 \text{ and }\lambda(x)^2 \ge m\|\Delta x_{\text{nt}}\|_2^2 \text{ and } \tilde f(t) = f(x + t\Delta x_{\text{nt}}) \text{ and } t \ge 0 $$ $$ \text{and } \lambda(x)^2 = \Delta x_{\text{nt}}^T \nabla f(x) \Delta x_{\text{nt}}$$
Integrating this inequality gives
$$\tilde f'(t) \le \tilde f'(0) + t\lambda(x)^2 + t^2\frac{L}{2m^{3/2}}\lambda(x)^3$$
where $\tilde f'(0) = - \lambda(x)^2$
Where did the $\tilde f'(0)$ come from? Shouldn't it just be the last two terms?