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PROOF 137

Prove.

At an extremum the derivative is 0

On an open interval.

What you may use

A1F ≤ G ⟹ lim F ≤ lim G
Limits preserve order (Proof 39)
A2lim c = c
Limit of a constant function
A3left limit = right limit ⟺ limit exists
One-sided limits (Proof 39)
A4at most one limit value
Uniqueness of the limit (Proof 39)
A5h > 0 ⟹ t/h has the same sign as t
Order and division (positive)
A6h < 0 ⟹ t/h has the opposite sign to t
Order and division (negative)
A7a ≤ b, b ≤ a ⟹ a = b
Antisymmetry of order
A8a ≤ b ⟺ a − b ≤ 0
Order and subtraction
A9f′− ≠ f′+ ⟹ not differentiable
One-sided derivatives (Proof 134)
A10a ≠ 0 ⟹ a·a⁻¹ = 1
Multiplicative inverse
A11(−1)a = −a
Signs and multiplication
A12−(−a) = a
Sign of a sign
A13(ab)c = a(bc)
Associativity of multiplication
A14a × 1 = a
Multiplicative identity
A15a + b = b + a
Commutativity of addition
A16a + (−a) = 0
Additive inverse
A17a + 0 = a
Additive identity
A180 ≤ |t|
Absolute value is nonnegative
A191 ≠ 0
1 and 0 are distinct numbers
A201 ≠ −1
1 and −1 are distinct numbers

Nothing outside this list may be cited.

Definition D1

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

Proof.

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