mean-value theorem

Let $f:ℝ\to ℝ$ be a function which is continuous on the interval $[a,b]$ and differentiable on $(a,b)$. Then there exists a number such that

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}.$$(1)

The geometrical meaning of this theorem is illustrated in the picture:

The dashed line connects the points $(a,f(a))$ and $(b,f(b))$. There is $c$ between $a$ and $b$ at which the tangent to $f$ has the same slope as the dashed line.

The mean-value theorem is often used in the integral context: There is a $c\in [a,b]$ such that

$$(b-a)f(c)={\int }_a^{b}f(x)𝑑x.$$(2)