Mean Value Theorem

Equivalent forms

f(b)-f(a) = f'(c)(b-a)

\text{Rolle: } f(a)=f(b) \Rightarrow f'(c)=0

The workhorse behind error bounds, Taylor's remainder, and the proof that a zero derivative means a constant function.

Symbol	Name	SI unit	Dimension	Range
$a$	Left endpoint Start of the interval	1	1	-3 – 0
$b$	Right endpoint End of the interval	1	1	0.5 – 3
$c$	Mean-value pointoutput Interior point where slopes match	1	1	-3 – 3
$m$	Secant slopeoutput Average rate of change over [a,b]	1	1	-5 – 5

Unit systems

Symbol	SI	CGS	Imperial
$a$	1	1	1

Where it holds

Requires f continuous on [a,b] and differentiable on (a,b). Fails for corners (|x|) or vertical tangents where f' doesn't exist.

Dimensional analysis

f'(c) (dim f / dim x)

(f(b)-f(a))/(b-a) (dim f / dim x)

Both are a difference in f divided by a difference in x — same dimension.

\checkmark

Discovery

Joseph-Louis Lagrange · 1797

Rolle stated the flat special case in 1691; Lagrange gave the general theorem in his 1797 Théorie des fonctions analytiques. Cauchy later made it rigorous. It is the bridge that turns local derivative information into global statements about functions.

Try this

Drive from A to B averaging 60 mph and, at some instant, your speedometer read exactly 60. The Mean Value Theorem guarantees that moment exists — no matter how you sped up and slowed down.

Move the endpoints of a secant line; somewhere between them a tangent runs exactly parallel to it. Watch the parallel-tangent point track the pair.

Research status: stable

Real-world applications

Average-speed traffic enforcement
Error bounds in numerical methods (Taylor remainder)
Proving monotonicity from the sign of f'
Establishing Lipschitz bounds for differential equations

Common misconceptions

The point c is unique — there may be several.
MVT gives you c explicitly — it only guarantees existence.
Continuity alone suffices — you also need differentiability on the open interval.

Experimental verification

A direct kinematic consequence: any journey has an instant where instantaneous speed equals average speed — confirmed by every speed-camera average-speed enforcement zone.

Derivation

Define

g(x)=f(x) - [f

(a) + m(x-a)] with m the secant slope; then g(a

)=g(b)=0

.

Rolle's theorem gives a point c with

g'(c)=0

, i.e.

f'(c)=m

.

Limiting cases

f(a

) = f(b)

⟶

f'(c) = 0

Rolle's theorem — a horizontal tangent exists inside.

b \to a

⟶

c \to a

The interval collapses onto the instantaneous derivative.

f linear⟶ c is anywhereA straight line has constant slope; every point satisfies the equality.

What if…

What if f has a corner in (a,b)?

Differentiability fails and the guarantee can break — |x| on [-1,1] has secant slope 0 but no interior point with zero derivative.

What if I need f(b)-f(a) bounded?

1

Find c for f(x)=x² on [0,2]

Given ·

f:: $x^{2}$
a:: 0
b:: 2

Find · c

Steps

Secant slope $= (4-0)/(2-0) = 2$
$f'(c) = 2c = 2$
$c = 1$

Result ·

c = 1

2

Speedometer guarantee

Given ·

avg:: 60 mph

Find · instant with

v = 60

Steps

Position is differentiable in time
Average velocity = secant slope
MVT: some instant matches it exactly

Result · such an instant must exist

Fundamental Theorem of Calculus→Taylor Expansion→l hopital ruleGreen's Theorem→