A graph is concave up on an open interval when its first derivative is increasing there, which happens when the second derivative is positive (f''(x) > 0). Visually, the curve bends upward like a cup, and every tangent line sits below the graph.
Concave up describes the shape of a curve, not its direction. On an open interval, a function is concave up when its first derivative is increasing on that interval, which is exactly when f''(x) > 0. Picture a cup or a smiley face. The function can be rising or falling while concave up; what matters is that its slope is getting bigger as you move right (from steeply negative, toward zero, toward steeply positive).
The AP framing comes straight from Topic 5.6: the second derivative tells you about concavity, and where concavity switches (from up to down or vice versa) you get an inflection point. One more geometric fact does a ton of work on the exam. When a curve is concave up, every tangent line lies below the curve, so a tangent line approximation underestimates the true function value (Topic 4.6).
Concave up lives mainly in Unit 5 (Analytical Applications of Differentiation), supporting LO 5.6.A and LO 5.9.A, both of which ask you to justify conclusions about a function from the behavior of its derivatives. That word "justify" is the whole game. On FRQs, "the graph looks curvy" earns zero points; "f''(x) > 0 on the interval, so f is concave up" earns the point. Concavity also reappears in Unit 4 (LO 4.6.A) to decide whether a linearization over- or underestimates, and on the BC exam in Unit 9 (LO 9.2.A), where you compute d²y/dx² for parametric curves by dividing d/dt(dy/dx) by dx/dt and then read off concavity the same way. One concept, three units.
Keep studying AP Calculus Unit 5
Visual cheatsheet
view galleryInflection Point (Unit 5)
An inflection point is where concavity flips, so concave up and inflection points are two halves of the same analysis. You find candidates where f'' = 0 or is undefined, then confirm that f'' actually changes sign there. No sign change, no inflection point.
Second Derivative Test (Unit 5)
Concavity is why the test works. If f'(c) = 0 and f''(c) > 0, the curve is concave up at c, so that critical point sits at the bottom of a cup. That makes it a local minimum.
Tangent Line Approximation (Unit 4)
On a concave up interval, tangent lines hug the curve from below, so any tangent line approximation is an underestimate. This is one of the most reliably tested concavity facts in Unit 4, and it shows up in FRQs like the 2017 cooling-potato problem.
Second Derivatives of Parametric Equations (Unit 9, BC only)
For parametric curves, you can't just differentiate y twice. You compute d²y/dx² as d/dt(dy/dx) divided by dx/dt, then check its sign exactly like in Unit 5. Positive means the parametric curve is concave up at that point.
Multiple choice loves combining concavity with direction. Stems like "a function has a negative first derivative and a positive second derivative" expect you to translate that into "decreasing and concave up" (falling, but leveling off). Other MCQs test whether a tangent line approximation is an underestimate, which is a concavity question in disguise.
On FRQs, concave up shows up constantly in justification parts. The 2018 FRQ Q3 gave you the graph of g = f' and asked you to reason about f, so concave up meant "g is increasing." The 2021 FRQ Q4 did the same with G(x) = ∫₀^x f(t) dt, where G is concave up wherever f (which is G') is increasing. The 2023 FRQ Q5 used a table of values, and the 2017 potato problem leaned on concavity to interpret a tangent line estimate. The scoring pattern is consistent. You earn the point by citing the sign of the second derivative (or the increasing behavior of the first derivative) on the interval, not by describing the picture.
Increasing is about the sign of f' (the function is going up). Concave up is about the sign of f'' (the slope is going up). They're independent. A function can be decreasing and concave up at the same time, like e^(-x), which falls forever but flattens out as it goes. Mixing these up is the single most common concavity error on the exam, so always ask: am I being asked about f' or about f''?
A function is concave up on an open interval exactly when its first derivative is increasing there, which is when f''(x) > 0.
Concave up describes shape, not direction, so a function can be decreasing and concave up at the same time (falling but flattening out).
On a concave up interval, every tangent line lies below the curve, so a tangent line approximation gives an underestimate.
Inflection points occur where concavity changes sign, and you must verify the sign change of f'', not just that f'' = 0.
If f'(c) = 0 and f''(c) > 0, the Second Derivative Test says f has a local minimum at c, because the curve is cupped upward there.
For BC parametric curves, find concavity from d²y/dx², computed by dividing d/dt(dy/dx) by dx/dt.
A graph is concave up on an interval when it bends upward like a cup, which happens exactly when the first derivative is increasing, i.e., when the second derivative is positive. It's tested in Topics 5.6 and 5.9 under LO 5.6.A and LO 5.9.A.
Yes, and this is a favorite MCQ trap. If f' < 0 and f'' > 0, the function is falling but its slope is increasing toward zero, so it levels off as it decreases. Think of e^(-x).
Increasing means f' > 0 (the y-values are going up). Concave up means f'' > 0 (the slope itself is going up). One is about the first derivative, the other is about the second, and any combination of the two is possible.
Neither, on its own. f'' = 0 only gives you a candidate for an inflection point. You need f'' to actually change sign there. For example, f(x) = x⁴ has f''(0) = 0 but is concave up everywhere with no inflection point at x = 0.
An underestimate. When the curve is concave up, the tangent line sits below the curve near the point of tangency, so the linearization comes out smaller than the true value. This is the Topic 4.6 connection the exam tests repeatedly.