A function is concave down on an interval when its derivative f' is decreasing there, which happens when the second derivative f'' is negative. The graph bends downward like a frown, and every tangent line on that interval sits above the curve.
Concave down describes how a graph bends, not whether it goes up or down. The CED definition (Topic 5.6) is precise: a function is concave down on an open interval if its derivative f' is decreasing on that interval. Since f'' tells you how f' is changing, that's equivalent to saying f''(x) < 0 there. Picture it as the slopes getting smaller as you move left to right. A function can be increasing and concave down at the same time (think of a rocket slowing down as it climbs), so don't confuse the bend with the direction.
The shape has a geometric consequence the exam loves. On a concave down interval, every tangent line lies above the curve. That's why tangent line approximations overestimate the true function value when the function is concave down (Topic 4.6). And wherever the graph switches from concave down to concave up (or vice versa), you've found an inflection point, which you locate by checking where f'' changes sign.
Concave down lives at the heart of Unit 5 (Analytical Applications of Differentiation), supporting learning objectives 5.6.A and 5.9.A, both of which ask you to justify conclusions about a function's behavior using its derivatives. The essential knowledge is direct. The graph of f is concave down on an open interval if f' is decreasing on that interval, and the second derivative tells you intervals of downward concavity and locates inflection points. It also powers Unit 4's linearization topic (4.6.A), where concavity determines whether a tangent line gives an overestimate or underestimate. For BC, Topic 9.2 extends the same idea to parametric curves, where you compute d²y/dx² by dividing d/dt(dy/dx) by dx/dt and check its sign. One concept, three units. That's why it shows up constantly on both the MCQ and FRQ sections.
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Visual cheatsheet
view galleryInflection Point (Unit 5)
An inflection point is where concavity flips, so it's where concave down ends or begins. Per the CED, you use the second derivative to locate inflection points by finding where f'' changes sign, not just where f'' = 0.
First Derivative (Unit 5)
Concave down is really a statement about f', not f. If you're handed the graph of f' (a classic FRQ setup), f is concave down exactly where the f' graph is going downhill. No second derivative needed.
Second Derivative Test (Unit 5)
Concave down is what makes the Second Derivative Test work for maxima. If f'(c) = 0 and f''(c) < 0, the curve frowns at c, so c must be a local maximum. The frown shape forces a peak.
Linearization and Tangent Line Approximation (Unit 4)
When a function is concave down near the point of tangency, the tangent line sits above the curve, so the tangent line approximation is an overestimate. This is the standard 'over or under?' question, and concavity is the one-word answer.
Second Derivatives of Parametric Equations (Unit 9, BC)
On BC, concavity questions show up for parametric curves. You compute d²y/dx² as d/dt(dy/dx) divided by dx/dt, then check its sign. Negative means concave down, same rule, fancier setup.
Multiple choice loves quick sign checks. "If the second derivative of a function is negative, what does it indicate?" and "When is the tangent line approximation an underestimate?" are both standard stems, and both come down to reading concavity correctly. On FRQs, concave down shows up in two recurring forms. First, the graph-of-f' problem: the 2023 BC FRQ Q4 gave the graph of f' (line segments and a semicircle) and asked you to reason about f, which means translating "f' is decreasing here" into "f is concave down here" and justifying it in writing. Second, table problems like 2023 FRQ Q5, where twice-differentiable functions and their derivative values at selected points set up concavity and approximation reasoning. The key skill is justification. "f is concave down on (a, b) because f''(x) < 0 on (a, b)" or "because f' is decreasing on (a, b)" earns the point. "The graph frowns" does not.
Decreasing is about direction (f' < 0, the graph goes downhill). Concave down is about bend (f'' < 0, the slopes are shrinking). They're independent. A function can be increasing and concave down at the same time, like a ball thrown upward that's still rising but slowing down. Mixing these up is one of the most common ways to lose justification points on FRQs.
A function is concave down on an interval exactly when f' is decreasing there, which is the same as f'' being negative on that interval.
Concave down does not mean decreasing; a function can be increasing and concave down at the same time, like an object rising but slowing down.
On a concave down interval, tangent lines lie above the curve, so a tangent line approximation gives an overestimate of the actual function value.
An inflection point occurs where the graph changes between concave up and concave down, found where f'' changes sign, not just where f'' equals zero.
If you're given the graph of f', the function f is concave down wherever the graph of f' is decreasing.
On BC, check concavity of a parametric curve by computing d²y/dx² as d/dt(dy/dx) divided by dx/dt and looking at its sign.
Concave down means the graph of f bends downward like a frown on an interval, which happens when f' is decreasing there, or equivalently when f''(x) < 0. It's the official CED definition from Topic 5.6.
No. Concave down describes the bend (f'' < 0), while decreasing describes the direction (f' < 0). A function like the height of a thrown ball on the way up is increasing but concave down because it's rising while slowing.
Overestimate. When f is concave down near the point of tangency, the tangent line lies above the curve, so the linearization value is larger than the actual function value. This is the standard Topic 4.6 question.
Concave down is a property of an interval; a local maximum is a single point. They connect through the Second Derivative Test, where f'(c) = 0 plus f''(c) < 0 (concave down at c) guarantees a local max at c.
Write that f''(x) < 0 on the interval, or that f' is decreasing on the interval, and name the interval. The 2023 BC FRQ Q4 rewarded exactly this reasoning from the graph of f'. Vague phrases like "the graph curves down" don't earn the justification point.