Theorems

Mean Value Theorem

Let

• and exists

Then such that:

Proof Hint

• Define in terms of , , ,
• Use single variable MVT for the terms

Schwarz’s theorem

Aka. Clairaut’s theorem on equality of mixed partials.

Let be open. Let . If and are continous on then .

Proof Hint

• Take a square neighbourhood around with side length .
• Define
• Use single variable MVT for , then
• Prove where is between and and is between and
• Rearrange the terms and prove where is between and and is between and
• Take the limit