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(** *           Local extremum's condition                  *)
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(* Specifications: *)
Theorem deriv_maximum :
forall f (a b c:R) (pr:derivable_pt f c),
a < c ->
c < b ->
(forall x:R, a < x -> x < b -> f x <= f c) -> derive_pt f c pr = 0.
Theorem deriv_minimum :
forall f (a b c:R) (pr:derivable_pt f c),
a < c ->
c < b ->
(forall x:R, a < x -> x < b -> f c <= f x) -> derive_pt f c pr = 0.
Theorem deriv_constant2 :
forall f (a b c:R) (pr:derivable_pt f c),
a < c ->
c < b -> (forall x:R, a < x -> x < b -> f x = f c) -> derive_pt f c pr = 0.
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Lemma nonneg_derivative_0 :
forall f (pr:derivable f),
increasing f -> forall x:R, 0 <= derive_pt f x (pr x).