5.6 Determining Concavity

Concavity tells you which way the graph is “bending.” Quick rules from the CED (FUN-4.A): - f is concave up on an open interval when f′ is increasing there (equivalently f″(x) > 0). - f is concave down when f′ is decreasing (equivalently f″(x) < 0). - A point of inflection is where concavity changes (f″ changes sign or f′ changes from increasing to decreasing)—check points where f″ = 0 or f″ is undefined (and confirm the sign change). How to decide in practice: 1. Compute f′ and f″. 2. Solve f″(x) = 0 and find where f″ is undefined; these partition the domain. 3. Make a sign chart for f″ on each interval (or look at whether f′ is increasing/decreasing). 4. State intervals where f″>0 (concave up) and f″<0 (concave down). Check endpoints/nondifferentiable points separately. This is exactly what AP wants: justify concavity using derivatives and a sign chart (see the Topic 5.6 study guide) (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V). For extra practice, try problems in the Unit 5 review (https://library.fiveable.me/ap-calculus/unit-5) or the 1000+ practice questions (https://library.fiveable.me/practice/ap-calculus).

Use the second derivative. For a twice-differentiable function f: - If f''(x) > 0 on an open interval, f is concave up there (tangent lines lie below the graph). - If f''(x) < 0 on an open interval, f is concave down there (tangent lines lie above the graph). - A point of inflection occurs where f'' changes sign (it may be where f'' = 0 or where f'' is undefined but f′ changes from increasing to decreasing or vice versa). Equivalent idea (CED FUN-4.A): f is concave up where f′ is increasing and concave down where f′ is decreasing. For AP problems, make a sign chart for f'' (include points where f'' is undefined), state intervals of concavity and any inflection points by checking sign changes, and justify with derivatives (FUN-4.A.4–6). For a quick review, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and more practice at (https://library.fiveable.me/practice/ap-calculus).

Use the first derivative when you want to know whether f is increasing or decreasing; use the second derivative (or whether f′ is increasing/decreasing) when you want concavity. Concretely: - Concave up on an open interval ⇔ f′ is increasing there ⇔ f″(x) > 0 on that interval (when f″ exists). - Concave down on an open interval ⇔ f′ is decreasing there ⇔ f″(x) < 0 on that interval. So: if you have f, compute f′ to find where f increases/decreases and critical points; then compute f″ (or look at the graph of f′) to determine concavity and locate possible inflection points (where f″ changes sign or f′ switches from increasing to decreasing). Remember to check points where f″ doesn’t exist but concavity can still change (point of nondifferentiability). This matches CED FUN-4.A (use f′ increasing/decreasing and f″ to find concavity/inflection). For AP prep, review Topic 5.6 on Fiveable (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Concave up means the graph bends like a cup that holds water (tangent lines lie below the curve); concave down means it bends like an upside-down cup (tangents lie above). Using the CED language: f is concave up on an interval when f′ is increasing there, and concave down when f′ is decreasing (FUN-4.A.4). Practically, check f′′: if f′′(x) > 0 the graph is concave up, if f′′(x) < 0 it’s concave down (FUN-4.A.5). Points of inflection occur where concavity changes (f′′ changes sign)—you may need to check points where f′′ = 0 or doesn’t exist and confirm the sign change (FUN-4.A.6). On the AP exam you’ll often justify concavity from f′ or f′′ graphs/tables, so practice making derivative sign charts. For a quick study refresher, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Step-by-step: 1. Find f′(x) and then f″(x). The second derivative tells you concavity (FUN-4.A.5). 2. Determine the domain of f and f″—include points where f or f″ is undefined (point of nondifferentiability). 3. Solve f″(x) = 0 and list where f″ is undefined. These partition the domain into open intervals. 4. Make a sign chart: pick one test x in each interval, evaluate the sign of f″(x). - If f″(x) > 0 on an interval, f is concave up there (derivative increasing). - If f″(x) < 0, f is concave down (derivative decreasing). (FUN-4.A.4) 5. Check endpoints where f″ = 0 or undefined: an inflection point occurs only where concavity actually changes sign and f is continuous there (FUN-4.A.6). On the AP exam you’ll justify using f″ and sign charts; practice this approach (topic study guide: https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V). For more problems, try the practice set (https://library.fiveable.me/practice/ap-calculus).

An inflection point is simply a point on f where the concavity changes—from concave up to concave down or vice versa. On the AP CED this is a change in whether f′ is increasing or decreasing (FUN-4.A.4). Practically: - Use f′′ when you can: if f′′(a) changes sign at x = a (positive → negative or negative → positive), then x = a is a point of inflection. - Candidates for inflection points come from where f′′(x) = 0 or f′′ is undefined; you must check that concavity actually changes (make a sign chart for f′′ or check whether f′ increases/decreases). - Graphically: f is concave up where f′ is increasing and concave down where f′ is decreasing (FUN-4.A.5). So an inflection corresponds to where the graph of f′ switches from rising to falling or vice versa. On the exam, state your candidate values, test sign changes for f′′ (or f′), and justify your conclusion—don’t just plug f′′=0. For a quick review, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and try problems from the Unit 5 practice set (https://library.fiveable.me/practice/ap-calculus).

The second-derivative test for concavity works whenever you can compute f'' on an open interval and use its sign there. Concretely: - If f''(x) > 0 for every x in an open interval, f′ is increasing there and f is concave up on that interval. - If f''(x) < 0 for every x in an open interval, f′ is decreasing there and f is concave down. Be careful: f''(x) = 0 (or undefined) at a point is only a candidate for a point of inflection—it’s not enough by itself. To confirm an inflection point you must show concavity actually changes sign (use a sign chart for f'' or check f′ switches from increasing to decreasing or vice versa). On the AP exam you should justify your conclusion (cite FUN-4.A.4–6): show where f'' is positive/negative or show f′ increasing/decreasing. For extra practice, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and try problems at (https://library.fiveable.me/practice/ap-calculus).

If f''(x) > 0 on an open interval, f′ is increasing there and the graph of f is concave up—it “cups” upward (tangent lines lie below the curve, the curve bends like a smile). If f''(x) < 0, f′ is decreasing and f is concave down—it “cups” downward (tangents lie above the curve, bends like a frown). Use this to find intervals of concavity by testing sign of f'' and to locate inflection points where f'' changes sign (and f is continuous/differentiable there)—see FUN-4.A.4–6 in the CED. Also, if f′(c)=0, the second-derivative test: f''(c)>0 → local minimum, f''(c)<0 → local maximum (if f''(c)=0 the test is inconclusive). For worked examples and AP-style practice on Topic 5.6, check the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Think of the derivative f′ as the slope of the tangent line at each x. Concavity describes how those slopes change as x moves. - If f′ is increasing on an interval, the slopes are getting larger—tangent lines tilt upward as you move right—so the graph of f bends upward (concave up). That corresponds to f″(x) > 0 there because f″ is the rate of change of f′. - If f′ is decreasing, slopes get smaller, the curve bends downward (concave down), and f″(x) < 0. So f″ tells you whether f′ is increasing or decreasing (FUN-4.A.4 and FUN-4.A.5). A point of inflection is where concavity changes sign—usually where f″ changes sign (FUN-4.A.6), although you must check f is continuous there (it could be a point of nondifferentiability). A quick graph check: look at the graph of f′—where it goes up, f is concave up; where it goes down, f is concave down. For extra practice and AP-aligned examples, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Find candidate x-values by solving f''(x) = 0 and where f'' is undefined. Those are possible inflection points, but not all are real ones. For each candidate, check whether concavity actually changes there: either - make a sign chart for f''(x) (pick test x-values left and right of the candidate) and confirm f'' changes from + to − or − to +, or - check f′(x): an inflection occurs where f′ switches from increasing to decreasing (or vice versa), since concave up ⇔ f′ increasing and concave down ⇔ f′ decreasing (CED FUN-4.A.4–5). If f'' = 0 but the sign of f'' does not change, there’s no inflection. Also remember points where f is not differentiable can be inflection points if concavity changes across them (CED FUN-4.A.6). On the exam, show work: list candidates, test intervals, and conclude intervals of concavity and any inflection points. For extra practice and worked examples, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and more practice at (https://library.fiveable.me/practice/ap-calculus).

Look for how the curve “bends” and how the slope (tangent) changes. - Concave up: the graph bends upward (like a cup). Tangent lines lie below the curve, and the slope f′ is increasing on that interval → f′ rising, so f″ > 0 (if f″ exists). - Concave down: the graph bends downward (like an upside-down cup). Tangents lie above the curve, f′ is decreasing → f″ < 0. - Inflection point: where concavity changes. Check where f″ = 0 or undefined and confirm f″ changes sign (or f′ switches from increasing to decreasing or vice versa). On exams you can use a derivative sign chart: find critical x where f″ = 0 or undefined, test nearby points to decide sign of f″ (FUN-4.A.4–6). If you’re given f′’s graph, just read where f′ increases/decreases to mark concavity. For practice and more examples, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V), the Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5), and extra problems (https://library.fiveable.me/practice/ap-calculus).

Step-by-step process to analyze concavity over the entire domain: 1. Find f′ and f″ (compute first and second derivatives). 2. Determine domain: where f, f′, and f″ are defined. Points where f′ or f″ are undefined or where f is not differentiable are potential inflection or endpoint issues. 3. Solve f″(x)=0 and find where f″ is undefined—these split the domain into test intervals. 4. Make a sign chart for f″ on each interval (pick one x-value per interval). - If f″(x)>0 on an interval, f is concave up there (f′ increasing). - If f″(x)<0, f is concave down there (f′ decreasing). 5. Identify points of inflection: points in the domain where f″ changes sign (must be in domain of f). If f″=0 but doesn’t change sign, it’s not an inflection point. 6. Summarize: list intervals of concave up/down and all inflection points; mention any nondifferentiable points separately. This matches CED FUN-4.A (use f″ to locate concavity and inflection). For extra practice and examples, check the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and thousands of practice problems (https://library.fiveable.me/practice/ap-calculus).

Because concavity describes how the slope (the first derivative) changes, concave up means the slope is getting larger as x increases. Formally: if f''(x) > 0 on an interval, then f′(x) is increasing on that interval (CED FUN-4.A.4–5). So when a graph is concave up the tangent lines tilt upward as you move right—slopes become less negative or more positive. That’s why textbooks say “the derivative is increasing” for concave up. Quick picture: if f′ goes from 1 to 3 across an interval, f is concave up there; if f′ goes from 3 down to 1, f is concave down (f″ < 0). Use f′ sign charts and f″ to justify concavity and find inflection points (FUN-4.A.5–6). For more practice and AP-style explanations, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and hundreds of practice problems (https://library.fiveable.me/practice/ap-calculus).

On AP free-response, handle concavity like a short proof: find f′ and f″, identify where f″ exists or fails to exist, and make a sign chart for f″ (or describe where f′ is increasing/decreasing). Use the CED language: say “f is concave up where f″>0, concave down where f″<0,” and mark any points where f″=0 or is undefined as candidate inflection points—then verify a change in concavity there. If you’re given f′ (graph or formula), state concavity from whether f′ is increasing (concave up) or decreasing (concave down). Always justify each interval with the sign of f″ or the behavior of f′; don’t just assert it. For endpoints or nondifferentiable points, note domain restrictions. On the FRQ, show the derivative work, the sign chart, and a sentence tying signs to concavity—that’s full credit style. For a focused review and practice problems, see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and more practice at (https://library.fiveable.me/practice/ap-calculus).

When f''(x) = 0 at a point, that point is a candidate for a change in concavity (a possible inflection point) but it’s not guaranteed to be one. Concavity actually depends on the sign of f' (increasing/decreasing) or f'': - If f'' changes from positive to negative at the point, f goes from concave up to concave down—that point is an inflection point. - If f'' changes from negative to positive, f goes from concave down to concave up—also an inflection point. - If f''(x) = 0 but f'' doesn’t change sign (stays ≥0 or ≤0), concavity doesn’t change and there’s no inflection point. Also check cases where f'' is undefined: an inflection can occur there if concavity changes. On the AP, justify with a sign chart for f'' (or show f' increasing/decreasing per FUN-4.A.4–6) rather than just citing f'' = 0. For a quick review see the Topic 5.6 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V) and the Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5). For more practice problems, try the AP practice bank (https://library.fiveable.me/practice/ap-calculus).