LMV Theorem for f in [1, 2]
\(\forall\,c \in\left(1, 2\right) \frac{f\left(2\right)-f\left(1\right)}{2-1}\)
\(=f'\left(c\right) \le 2\)
\(f\left(2\right)-f\left(1\right) \le 2\)
\(\Rightarrow f\left(2\right) \le 4 \quad...(1)\)
Again, using LMV Theorem in \(\left[2, 4\right]\)
\(\forall\,d \in\left(2, 4\right) \frac{f\left(4\right)-f\left(2\right)}{4-2}\)
\(=f'\left(d\right) \le 2\)
\(\therefore f\left(4\right)-f\left(2\right) \le 4\)
\(\Rightarrow 8-f\left(2\right) \le 4\)
\(\Rightarrow 4 \le f\left(2\right)\)
\(\Rightarrow f\left(2\right) \ge 4\)
From \(\left(1\right) and \left(2\right), we\ get \ f\left(2\right)=4\ LMV\ Theorem \ for \ f \ in \ [1, 2]\)
\(\forall\,c \in\left(1, 2\right) \frac{f\left(2\right)-f\left(1\right)}{2-1}\)
\(=f'\left(c\right) \le 2\)
\(f\left(2\right)-f\left(1\right) \le 2\)
\(\Rightarrow f\left(2\right) \le 4 \quad...(1)\)
Again, using LMV Theorem in \(\left[2, 4\right]\)
\(\forall\,d \in\left(2, 4\right) \frac{f\left(4\right)-f\left(2\right)}{4-2}\)
$$=f'\left(d\right) \le 2$$
$$\therefore f\left(4\right)-f\left(2\right) \le 4$$
$$\Rightarrow 8-f\left(2\right) \le 4$$
$$\Rightarrow 4 \le f\left(2\right)$$
$$\Rightarrow f\left(2\right) \ge 4 \quad ...(2)$$
From $$\left(1\right)$$ and $$\left(2\right)$$, we get $$f\left(2\right)=4$$