A relative (local) minimum is a point where a function's value is lower than at all nearby points, even if it isn't the lowest value on the whole graph. On the AP Calculus exam, you locate one where f' changes from negative to positive, or where f'(c) = 0 and f''(c) > 0.
A relative minimum (the CED also calls it a local minimum) is a point where a function dips down to a low value compared to everything around it. Picture a valley in a mountain range. It's the bottom of that valley, but there might be a deeper valley somewhere else on the graph. That deeper one would be the absolute minimum.
In AP Calculus, the definition is less important than the detection method. Relative minima can only happen at critical points, places where f'(x) = 0 or f' is undefined. From there, two tests confirm it. The First Derivative Test says f has a relative minimum at x = c if f' changes sign from negative to positive there (the function stops falling and starts rising). The Second Derivative Test says if f'(c) = 0 and f''(c) > 0, the curve is concave up at c, so c is a relative minimum. Either test works, but on the exam your justification has to name the behavior of the derivative, not just say "the graph goes down then up."
Relative minima live at the heart of Unit 5 (Analytical Applications of Differentiation), specifically Topics 5.4, 5.7, and 5.9. All three share the same learning objective, justifying conclusions about a function's behavior based on its derivatives (5.4.A, 5.7.A, 5.9.A). Essential knowledge FUN-4.A.2 states it directly. The first derivative determines the location of relative extrema. The concept then resurfaces in Unit 6 through accumulation functions (Topic 6.5, FUN-5.A.3), where you find minima of g(x) = ∫ₐˣ f(t) dt by reading the graph of f, which IS g'. There's also a high-value bonus rule from 5.7. If a continuous function has only one critical point on an interval and it's a relative minimum, it's automatically the absolute minimum there. That shortcut shows up constantly in justifications.
Keep studying AP Calculus Unit 5
Visual cheatsheet
view galleryFirst Derivative Test (Unit 5)
This is the main tool for finding relative minima. If f' changes sign from negative to positive at a critical point, f has a relative minimum there. The function was decreasing, then it started increasing, so it bottomed out.
Critical Points (Unit 5)
Critical points are your candidate list. Every relative minimum sits at a critical point, but not every critical point is a relative minimum (some are maxima, some are neither). You always find critical points first, then test them.
Absolute Minimum (Unit 5)
A relative minimum is the lowest point in its neighborhood; an absolute minimum is the lowest point on the entire interval. The CED gives you a bridge between them. With only one critical point on an interval, a relative minimum there is also the absolute minimum.
Accumulation Functions (Unit 6)
When g(x) = ∫ₐˣ f(t) dt, the graph of f is literally the graph of g'. So g has a relative minimum wherever f crosses from negative to positive. This is the same First Derivative Test wearing an integral costume, and it's one of the most common FRQ setups.
Relative minima appear in both multiple choice and FRQs, almost always as a "find it AND justify it" task. The classic FRQ setup gives you the graph of f' (not f) and asks where f has a relative minimum. The 2023 FRQ Q4 did exactly this, defining f on [−2, 8] with a graph of f' made of line segments and a semicircle. Your job is to find where f' changes from negative to positive and say so explicitly. MCQs love the Second Derivative Test, asking what f''(c) > 0 tells you at a critical point, or what happens when f''(c) = 0 (answer: the test is inconclusive, fall back to the First Derivative Test). Unit 6 versions hand you g(x) = ∫ₐˣ f(t) dt and test whether you recognize that f plays the role of g'. The scoring rubrics reward precise language. Write "f has a relative minimum at x = k because f' changes from negative to positive at x = k," not "the graph turns around."
A relative minimum only has to beat its neighbors; an absolute minimum has to beat every point on the interval. A graph can have several relative minima but at most one absolute minimum value. The exception worth memorizing comes from Topic 5.7. If a continuous function has exactly one critical point on an interval and it's a relative minimum, that point is also the absolute minimum. On FRQs, finding an absolute minimum usually requires the Candidates Test (compare critical points and endpoints), while a relative minimum only needs a sign change in f'.
A relative minimum is a point where the function's value is lower than at all nearby points, but it may not be the lowest value on the entire graph.
Relative minima can only occur at critical points, where f'(x) = 0 or f' is undefined.
The First Derivative Test confirms a relative minimum when f' changes sign from negative to positive at a critical point.
The Second Derivative Test confirms a relative minimum when f'(c) = 0 and f''(c) > 0; if f''(c) = 0, the test is inconclusive and you need the First Derivative Test instead.
If a continuous function has only one critical point on an interval and it's a relative minimum, it is automatically the absolute minimum on that interval.
For an accumulation function g(x) = ∫ₐˣ f(t) dt, the integrand f acts as g', so g has a relative minimum wherever f crosses from negative to positive.
A relative minimum is a point where a function's value is lower than at all nearby points, like the bottom of a valley. On the AP exam you find one at a critical point where f' changes from negative to positive, or where f'(c) = 0 and f''(c) > 0.
Yes, they're identical. The CED uses both names, writing "relative (local) extrema," so any test question using either term means the same thing.
A relative minimum is the lowest point compared to its neighbors, while an absolute minimum is the lowest point on the entire interval. One special case from Topic 5.7: if a continuous function has only one critical point on an interval and it's a relative minimum, it's also the absolute minimum there.
Not necessarily. When the second derivative is zero at a critical point, the Second Derivative Test is inconclusive, so the point could be a minimum, a maximum, or neither. Check the sign change of f' instead.
Write that f' changes sign from negative to positive at that x-value, or that f'(c) = 0 and f''(c) > 0. Rubrics require you to cite the derivative's behavior. Saying "the graph goes down then up" without mentioning f' won't earn the justification point.