Theorems

Mean Value Theorem

Let . Suppose and exists.

If then such that:

Proof Hint

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

Clairaut

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