Calculus I › Applications of derivatives › full formula sheet
Concavity
The second derivative reads the curve’s bend — smile or frown.
Concave up (∪, a smile — holds water) vs. concave down (∩, a frown — spills water). Bend is independent of uphill/downhill.
Before this lesson: Higher-order derivatives
Where it comes from
The problem: where does f(x) = x³ − 3x bend upward, and where downward? The naive guess — “concave up means increasing” — confuses bend with direction. Exhibit A, f(x) = −x² on (−∞, 0):
The right intuition: read the slopes left to right. On a smile ∪, slopes go −, 0, + — increasing. On a frown ∩, slopes go +, 0, − — decreasing. And “f′ increasing” is exactly what f″ > 0 says. That is the whole dictionary:
Before reading on: the slopes of f keep increasing as x grows. Does the graph cup upward or downward? Draw three tangent segments and see.
Derivation
Apply the increasing/decreasing theorem — not to f, but to f′.
Why “holds water”? A concave-up cup ∪ can hold water; concave-down ∩ spills it. And since tangents sit below a smile, a concave-up function always lies above its tangent lines — which is exactly why linear approximation underestimates concave-up functions (remember √x? No — that one bends down. Try x²: tangent at 1 is 2x−1, and x² ≥ 2x−1 always ✓).
Before reading on: f(x) = x⁴ has f″(0) = 0. Is (0, 0) an inflection point? What extra check decides it?
How to use it
The procedure — a carbon copy of the increasing/decreasing chart, one derivative higher:
- Compute f″.
- Find where f″ = 0 or is undefined (with f defined) — candidates where the bend might change.
- Split the domain at those points into intervals.
- Test the sign of f″ on each interval.
- Conclude: f″ > 0 ⇒ concave up; f″ < 0 ⇒ concave down — interval by interval.
Bend ≠ direction ≠ height
Three independent dials: f says where the graph is, f′ says which way it tilts, f″ says how it bends. All four combos exist — x² on (−∞, 0) is concave up and decreasing.
When to reach for it
Curve sketching (the bend completes the picture), finding inflection points (next page — where the bend flips), and the second derivative test (classifying maxima/minima by bend).
Worked examples
Four bends, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — the basic move: f(x) = x³
- Differentiate twice. f′(x) = 3x², f″(x) = 6x.
- Candidates. f″ = 0 at x = 0.
- Signs. 6x < 0 for x < 0; > 0 for x > 0.
- Conclude. Concave down on (−∞, 0), concave up on (0, ∞). (The cube bends down, flattens, bends up — an S.)
Your turn: f(x) = −x³ — where is it concave up / down?
Answer: Concave up on (−∞, 0); concave down on (0, ∞).
f″(x) = −6x: > 0 for x < 0 (up), < 0 for x > 0 (down) ✓.
Example 2 — flat but still up: f(x) = x4
- Differentiate twice. f′(x) = 4x³, f″(x) = 12x².
- Candidates. f″ = 0 at x = 0.
- Signs. 12x² ≥ 0 everywhere — and > 0 except at 0 itself.
- Conclude. Concave up on (−∞, 0) and on (0, ∞) — hence concave up everywhere. A zero of f″ with no sign change changes nothing.
Your turn: f(x) = ex — concavity?
Answer: Concave up everywhere.
f″(x) = ex > 0 for all x — no candidates, no flips.
Example 3 — with a domain break: f(x) = 1/x
- Differentiate twice. f′(x) = −x−2, f″(x) = 2x−3 = 2/x³.
- Candidates. f″ is never 0; undefined at x = 0 — but f is undefined there too, so it’s a domain break, not a candidate.
- Signs. 2/x³ < 0 for x < 0 (negative denominator); > 0 for x > 0.
- Conclude. Concave down on (−∞, 0), concave up on (0, ∞) — separately, never glued.
Your turn: f(x) = ln x (x > 0) — concavity?
Answer: Concave down on (0, ∞).
f′(x) = 1/x, f″(x) = −1/x² < 0 for every x > 0 ✓.
Example 4 — trigonometry: f(x) = sin x on [0, 2π]
- Differentiate twice. f′(x) = cos x, f″(x) = −sin x.
- Candidates. −sin x = 0 at x = 0, π, 2π.
- Signs. On (0, π): sin x > 0 → f″ < 0. On (π, 2π): sin x < 0 → f″ > 0.
- Conclude. Concave down on (0, π) (the arch ∩), concave up on (π, 2π) (the bowl ∪).
Your turn: f(x) = cos x on [0, 2π] — concavity intervals?
Answer: Concave up on (π/2, 3π/2); concave down on (0, π/2) and (3π/2, 2π).
f″(x) = −cos x > 0 exactly when cos x < 0, i.e. on (π/2, 3π/2) ✓.
Memorization tips
- Smile ∪ = up = f″ > 0. Draw the smile next to the formula until it’s automatic.
- Holds water / spills water: ∪ holds, ∩ spills. Say which while pointing at the graph.
- Slopes increasing: read tangent tilts left to right — −, 0, + is a smile. That image is the proof.
- Three dials: f = where, f′ = which way, f″ = how bent. Quiz yourself: “x² on (−∞, 0): up or down? increasing or decreasing?” (Up, decreasing.)
- Same chart, one derivative higher: the procedure is identical to increasing/decreasing — just applied to f″.
- Zero splits, sign decides: f″ = 0 only fences intervals; the sign on each side is the answer.
Final challenge
Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and concavity is yours.
← Back to the Calculus I formula sheet
How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
What is concavity?
Concavity is which way a curve bends: concave up (∪, like a smile — holds water) where f″ > 0, concave down (∩, like a frown) where f″ < 0.
Does concave up mean increasing?
No — bend and direction are independent. f(x) = x² is concave up everywhere but decreasing on (−∞, 0). Concavity reads the bend (f″); increasing reads the slope (f′).
Why does f″ > 0 mean concave up?
f″ > 0 means f′ is increasing — slopes grow left to right (−, 0, +). Slopes increasing through zero is exactly the smile shape ∪, bending upward.
What does f″ = 0 tell you about concavity?
By itself, nothing — it’s a candidate where concavity might change. f(x) = x4 has f″(0) = 0 but stays concave up; only a sign change of f″ marks a real bend change (an inflection point).
Where do tangent lines sit relative to the curve?
On a concave-up stretch, tangent lines lie below the curve (the smile sits above its tangents); on concave-down, tangents lie above. That’s why concave-up functions lie above their linear approximations.
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