PROOF 44

Prove.

The mean value theorem

On a closed interval.

What you may use

A1continuous on a closed interval ⟹ max and min
extreme value theorem (completeness)
A2F ≤ G ⟹ lim F ≤ lim G
limits preserve order (Proof 39)
A3lim c = c
limit of a constant function
A4left limit = right limit ⟺ limit exists
one-sided limits (Proof 39)
A5a limit has at most one value
uniqueness of limits (Proof 39)
A6h > 0 ⟹ t/h has the same sign as t
order and division (positive)
A7h < 0 ⟹ t/h has the opposite sign to t
order and division (negative)
A8a ≤ b, b ≤ a ⟹ a = b
antisymmetry of order
A9a < b ⟹ b − a ≠ 0
order and subtraction
A10(f ± g)′ = f′ ± g′
derivative of sums and differences (Proof 39)
A11(px+q)′ = p, px+q is continuous
linear functions
A12f, g continuous ⟹ f ± g continuous
sums and differences of continuous functions
A13a × 0 = 0
multiplication by 0 gives 0
A14a ≠ 0 ⟹ a·a⁻¹ = 1
multiplicative inverse
A15(ab)c = a(bc)
associativity of multiplication
A16ab = ba
commutativity of multiplication
A17a × 1 = a
multiplicative identity
A18a + (−a) = 0
additive inverse
A19a + 0 = 0 + a = a
additive identity
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

f is differentiable at x if the limit lim (h→0) [f(x+h) − f(x)]/h exists, and its value is written f′(x).

Proof.

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