Conditions
The function f(x) is increasing on an open interval.
f'(x) > 0
Any point in the domain of f(x) where f'(x) = 0 or f'(x) is undefined.
What is a critical point (or critical number)?
An x-intercept on the graph of f'(x) where the graph crosses from above the x-axis to below it indicates this feature on f(x).
What is a relative maximum?
The reason why writing "f(x) has a max because f'(x) changes signs" loses points on the AP Exam.
What is failing to specify the direction of the sign change (positive to negative)?
In Jujutsu Kaisen, this primary grade 1 sorcerer quit corporate life to return to sorcery because both are pain, but sorcery pays better.
Who is Kento Nanami?
The function f(x) is concave down on an open interval.
What is f''(x) < 0 (or f'(x) is decreasing)?
Guarantees that a continuous function on a closed interval [a, b] attains both an absolute maximum and an absolute minimum.
What is the Extreme Value Theorem (EVT)?
An interval where the graph of f'(x) is sloping downward indicates this concave behavior on f(x).
What is concave down?
The reason why f''(c) = 0 alone is not sufficient to claim x = c is an inflection point.
What is f''(x) must actually change signs at c?
This Hashira wears a mismatched haori in memory of his late friend Sabito and his sister Tsutako.
Who is Giyu Tomioka?
f(x) has a point of inflection at x = c.
What is f''(x) changes sign at x = c (or f'(x) changes from increasing to decreasing / decreasing to increasing)?
Requires f(x) to be continuous on [a, b] and differentiable on (a, b) to guarantee at least one value c where
f'(c)=(f(b)-f(a))/(b-a)
What is the Mean Value Theorem (MVT)?
A local peak or valley on the graph of f'(x) corresponds to this point on f(x).
What is a point of inflection?
The pronoun you should never use in an AP Calculus response when referring to a function or its derivative.
What is "it"?
This Public Safety Devil Hunter has contracts with both the Fox Devil and the Curse Devil, and uses a sword with a needle-like tip.
Who is Aki Hayakawa?
According to the First Derivative Test, f(x) has a relative minimum at x = c.
What is f'(x) changes from negative to positive at x = c?
A special case of the Mean Value Theorem where f(a) = f(b) = 0, guaranteeing a point where f'(c) = 0.
What is Rolle's Theorem?
If the graph of f''(x) is above the x-axis, then the graph of f'(x) must be doing this.
What is increasing?
The two conditions that must be checked before using the Mean Value Theorem on an interval [a, b].
What are continuity on [a, b] and differentiability on (a, b)?
Early in The Apothecary Diaries, Maomao solves the mysterious illness affecting the concubines' infant children by realizing their face powder contains this toxic heavy metal.
What is lead?
According to the Second Derivative Test, f(x) has a relative maximum at x = c.
What is f'(c) = 0 and f''(c) < 0?
The systematic process used to test critical points and endpoints on a closed interval to find absolute extrema.
What is the Candidates Test?
If f'(c) = 0 and the graph of f'(x) is strictly increasing through x = c, f(x) has this feature at c.
What is a relative minimum (and a point of inflection)?
The Second Derivative Test fails and yields no conclusion when this occurs at a critical point x = c.
What is f''(c) = 0 (or f''(c) is undefined)?
In Frieren: Beyond Journey's End, this stoic, highly skilled human mage is Fern's opponent during the Second Class Mage Exam and is known for her specialization in binding mages with hair.
Who is Übel?