See Darboux's theorem for notation, formulation, and significance.

Suppose we're given an intermediate value *y*. We must show *f'(c)=y* for some *a<=c<=b*. Define *g(x)=f(x)-y*x*. Then *g* is also a differentiable function, and *g'(x)=f'(x)-y*. If we assume *f'(a)<f'(b)* (the other case is absolutely symmetrical), then this means *g'(a)<0<g'(b)*.

So we know that *g* is locally decreasing in a right neighbourhood of *a* and locally increasing in a left neighbourhood of *b*. In particular, neither *a* nor *b* can be a minimum of *g*. But since the interval [*a,b*] is closed and bounded, we know that *g* attains a minimum on it, at a point we shall call *c*. Since *c* is in (*a,b*), we know that *g'(c)=0*, or *f'(c)=y*.

**QED.**