How to show if $\nabla^2 f(x) \succeq \alpha I$, then the function is $\alpha$-strongly convex?
In my optimization notes I have $$\nabla^2 f(x) \succeq \alpha I \rightarrow \alpha\text{-strongly convex} \,\,\,\,\,\,\forall x$$ where $x \in \mathbb{R}^n$ and $A\succeq B$ means $A-B$ is positive semi-definite.
For the proof we use mean value theorem $$ \nabla f(y)-\nabla f(y) = \int_0^1 \nabla^2 f(x+t(y-x))(y-x)dt $$ where $x, y \in \mathbb{R}^n$ and $0 \leq t \leq 1$. $$ \langle \nabla f(y)-\nabla f(y) ,y-x \rangle= \langle \int_0^1 \nabla^2 f(x+t(y-x))(y-x)dt,y-x \rangle $$ Since $$\nabla^2 f(x+t(y-x)) \succeq \alpha I \tag{1}$$ and the fact that $\langle Ad,d \rangle \geq c\|d\|^2 \leftrightarrow A \succeq cI \tag{2}$
$$ \langle \int_0^1 \nabla^2 f(x+t(y-x))(y-x)dt,y-x \rangle \geq \alpha \|y-x\|^2 \tag{3} $$ I do not understand why how we can use $(1)$ to get $(2)$? Because I do not know why we can use $(1)$ and write $$ A=\int_0^1 \nabla^2 f(x+t(y-x))dt \succeq \int_0^1 \alpha I dt= \alpha I $$ and then use $(2)$.
Is it possible to take integral from any matrix inequality?
In your statement of the "mean value theorem" and in other parts of your post, you are taking the integral of a vector or a matrix, which does not make sense.
I think it would be better to state the integral equality as $$\langle \nabla f(y) - \nabla f(x), y-x \rangle = \int_0^1 \langle \nabla^2 f(x + t(y-x)) (y-x), y-x \rangle \, dt.$$ (This is simply the fundamental theorem of calculus $g(1) - g(0) = \int_0^1 g'(t) \, dt$ applied to $g(t) := \langle \nabla f(x + t(y-x)), y-x \rangle$.)
From here, you can immediately use $\nabla^2 f(x + t(y-x)) (y-x), y-x \rangle \ge \alpha \|y-x\|^2$ from (2), as desired.