Mean Value Theorems

Geometrically

Geometrically, dy/dx or f'(x) represents the slope of the tangent to the curve y=f(x) at the point (x,y). If f'(x)=0, the tangent is parallel to the x

If f

If f:[a,b]→R is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c∈(a,b) such that f'(c)=0.

If f

If f:[a,b]→R is continuous on [a,b] and differentiable on (a,b), then there exists c∈(a,b) such that f'(c)=(f(b)-f(a))/(b-a).

There is at least one point

There is at least one point on the curve where the tangent is parallel to the secant AB joining the endpoints A(a,f(a)) and B(b,f(b)).

When f(a)=f(b)

When f(a)=f(b), the LMV formula becomes f'(c)=0, so Rolle's theorem is a special case of LMV theorem.