840. Does the function $f(x)=x^{2}+3$ satisfy the conditions of Lagrange's theorem on the segment $[-1,2]$? If the theorem is applicable, then find the point $c$ where $f(b)-f(a)=f^{\prime}(c)(b-a)$, where $a=$ $=-1, b=2$.
Solution. The function satisfies the conditions of Lagrange's theorem, as it is continuous on the segment $[-1,2]$ and has
a finite derivative in the interval ( $-1,2$ ). The point $c$ will be found from the equation

$$
f(2)-f(-1)=f^{\prime}(c)[2-(-1)]
$$

hence

$$
7-4=2 c \cdot 3, c=\frac{1}{2}
$$