1. Mean Value Theorems

Calculus 2



Weiqi Zhou

Extrema

  • if is defined in some neighbourhood of ，and holds for any ，then is a local maximum of

  • if is defined in some neighbourhood of ，and holds for any ，then is a local minimum of

  • local maximum and local minimum are also called local extrema (singular form: extremum)

Stationary Point

if ，then is a stationary point of

Differentiable Extrema Are Stationary

if is differentiable at ，and is an extremum

then is also stationary

Proof

  • suppose that is differentiable at an extremum



  • the denominator has opposite signs if we take and respectively

  • the numerator stays unchanged as is an extrmum, the limit has to be for it to exist

Stationary Points Are Not Necessarily Extrema

example:

The Rolle Theorem

if is continuous on ，differentiable in ，and , then there exists such that

Proof

  • being continuous on implies existences of global extrema

  • if is constant on ，then the conclusion is obvious

  • if is not constant on ，then implies that at least one of extrema is attained in

  • such a point is stationary

Example

  • show that has precisely one real root ()

  • is a real polynomial of order , it has at least one real root. If it has a triple root, then we shall be able to write as ，which is impossible

  • if it has distinct real roots, then by the Rolle Theorem, admits at least one real root between two neighbouring roots of , but has no real roots

Exercise

show that if an order polynomial admits distinct real roots，then its -th derivative () has distinct real roots

The Mean Value Theorem (Lagrange)

if is continuous on , and differentiable in ，then there exists so that

Proof

  • let

  • then is continuous on ，differentiable in , and

  • invoke the Rolle theorem to assert the existence of ，i.e.,

Example

  • show that if is differentiable on ，and ，then is constant

  • take arbitrary with

  • invoke the MVT to assert the existence of

  • but everywhere, which means

Exercises

  • show that if ， are both differentiable on ，and , then ，where is a constant

  • show that if
    (Hint: Apply MVT on in )

Exercises

  • show that if is differentiable，and ，，then

  • show that if is differentiable，and , then need not exist

The Mean Value Theorem (Cauchy)

if are continuous on ，differentiable on ，and has no zero on , then there exists

Proof

  • implies ，otherwise it contradicts the Rolle theorem

  • let

  • then is continuous on ，differentiable in ，and . Invoke the Rolle theorem to assert the existence of ，i.e.,

Typicial Mistake

one shall not try to apply the Lagrange MVT on and simultaneously to prove the Cauchy MVT, as and need not be the same

Summary

  • Extrema and Stationary Points

  • The Rolle Theorem

  • MVT