Question 7.T.10: Let c be a critical point and a point of continuity for the ...
Let c be a critical point and a point of continuity for the function f : D → \mathbb{R}.
(i) If there is an open interval I ⊆ D, which contains c, such that
f^{′}(x) < 0 for all x ∈ I, x < c
f^{′}(x) > 0 for all x ∈ I, x > c,
then f(c) is a local minimum for f.
(ii) If there is an open interval I ⊆ D, which contains c, such that
f^{′}(x) > 0 for all x ∈ I, x < c
f^{′}(x) < 0 for all x ∈ I, x > c,
then f(c) is a local maximum for f.
(iii) If there is an open interval I ⊆ D, which contains c, such that f^{′}(x) has the same sign on I\{c}, then f(c) is not a local extremum for f.
Figure 7.7 illustrates the three cases of the theorem. To prove case (i), note that, by Theorem 7.9, f is decreasing on the interval I ∩ (−∞, c] and increasing on the interval I ∩ [c, ∞), hence
f(x) ≥ f(c) for all x ∈ I, x < c,
f(x) ≥ f(c) for all x ∈ I, x > c.
Thus f(x) ≥ f(c) for all x ∈ I, which means f(c) is a local minimum.
The other two cases can be proved by similar arguments.