Concavity, Second Derivatives, and Inflection Points in Curve Sketching

Concavity as “How the Slope Is Changing”

When you already know whether a function is increasing or decreasing, concavity tells you something different: whether the slope is getting bigger or getting smaller as you move left to right. This is what makes concavity so useful for describing acceleration, diminishing returns, and the “shape” of a graph.

Concave up (cup-shaped)

• Plain-language definition: slopes are increasing as x increases.
• If the function is increasing, it increases faster and faster.
• If the function is decreasing, it decreases less and less steeply (it “levels off” while going down).

Concave down (cap-shaped)

• Plain-language definition: slopes are decreasing as x increases.
• If the function is increasing, it increases slower and slower (diminishing returns).
• If the function is decreasing, it decreases more and more steeply.

Quick mental picture: Imagine driving along the graph from left to right and watching the tangent line rotate. If the tangent line rotates upward (slope rising), the graph is concave up. If it rotates downward (slope falling), the graph is concave down.

Second Derivative Connection (and Common Misconceptions)

The first derivative f'(x) measures slope. The second derivative f''(x) measures how that slope changes.

Misconception: “If f''(c)=0, then c is an inflection point.”

Not necessarily. f''(c)=0 (or f'' undefined) is only a candidate for an inflection point. An inflection point occurs only when concavity actually changes (from up to down or down to up) as you pass through that x-value.

Example where f''(0)=0 but no inflection: f(x)=x^4. Then f''(x)=12x^2, so f''(0)=0, but f''(x)>0 for all x≠0. The graph is concave up on both sides, so there is no inflection point at x=0.

Continue in our app.

• Listen to the audio with the screen off.
• Earn a certificate upon completion.
• Over 5000 courses for you to explore!

Or continue reading below...

Download the app

Example where f'' is undefined at an inflection: f(x)=x^{1/3}. Then f''(x) is undefined at 0, but the concavity changes across 0, so x=0 is an inflection point.

Finding Intervals of Concavity with a Sign Chart for f''(x)

The most reliable method is: compute f''(x), find where it is zero or undefined, then test the sign of f'' on the resulting intervals.

Step-by-step procedure

• Step 1: Compute f''(x).
• Step 2: Solve f''(x)=0 and also note where f''(x) is undefined (but f is defined). These are possible inflection x-values.
• Step 3: Put these x-values in order on a number line to create test intervals.
• Step 4: Pick a test point in each interval and evaluate the sign of f'' there.
• Step 5: Conclude concave up where f''>0 and concave down where f''<0.

Worked example: sign chart for concavity

Let f(x)=x^3-3x. Find concavity and inflection points.

• Compute derivatives: f'(x)=3x^2-3, f''(x)=6x.
• Candidates: f''(x)=0 gives 6x=0 so x=0.
• Sign chart: test x=-1 and x=1.

Interval:     (-∞, 0)      (0, ∞)  Test x:         -1           1  f''(x)=6x:      -            +  Concavity:   down         up

So the graph is concave down on (-∞,0) and concave up on (0,∞).

Confirming Inflection Points Correctly (Sign Change Test)

An inflection point occurs at x=c if:

• f is continuous at c (in typical curve sketching problems), and
• the concavity changes across c, meaning f'' changes sign from positive to negative or negative to positive.

In the previous example, f''(x)=6x changes sign at 0, so x=0 is an inflection point. The inflection point is the coordinate (0,f(0))=(0,0).

Checklist: “Is it really an inflection point?”

• Did you identify c from f''(c)=0 or f'' undefined?
• Did you test the sign of f'' on both sides of c?
• Did the sign actually change?
• Is the point (c,f(c)) on the graph (i.e., does f(c) exist)?

Example: candidate that fails the sign-change test

Let f(x)=x^4. Then f''(x)=12x^2.

• f''(0)=0 gives a candidate x=0.
• But 12x^2 is positive for x<0 and for x>0.
• No sign change, so no inflection point.

Mini-Lab 1: Read Concavity from “Slope Behavior”

Goal: Practice describing a graph using two layers of information: (i) increasing/decreasing and (ii) concavity.

Instructions

• Sketch a smooth curve that has an S-shape (like a cubic), or use a provided graph from your notes.
• Mark three x-regions: left of the “middle bend,” around the bend, and right of the bend.
• On each region, annotate:

• Increasing or decreasing? (based on whether the graph goes up or down as x increases)
• Concave up or concave down? (based on whether slopes are increasing or decreasing)

Sentence frames (fill in with your observations)

• “From x=a to x=b, the function is increasing/decreasing and concave up/down, meaning the slope is getting steeper/less steep.”
• “Near x=c, the graph changes from concave ___ to concave ___, so x=c is/is not an inflection point.”

Mini-Lab 2: Build a Concavity Sign Chart from a Formula

Goal: Turn f''(x) into a clear concavity description.

Use g(x)=x^4-4x^3.

• Step 1: Compute g''(x).
• Step 2: Solve g''(x)=0 to get candidate x-values.
• Step 3: Make a sign chart for g'' and label concavity on each interval.
• Step 4: Identify which candidates are true inflection points (sign change required).

Hint (to self-check): If you factor g''(x), the sign analysis becomes much easier.

Why Concavity Communicates Acceleration and Diminishing Returns

Motion interpretation

If s(t) is position, then s'(t) is velocity and s''(t) is acceleration.

• Concave up in s(t): s''(t)>0, velocity is increasing (speeding up in the positive direction, or slowing down if moving negatively).
• Concave down in s(t): s''(t)<0, velocity is decreasing.

This is why concavity is often described as “acceleration” in a graph: it tells you whether the slope (velocity) is rising or falling.

Economics interpretation: profit and diminishing returns

Let P(x) be profit as a function of units produced x. Then:

• P'(x) is marginal profit (how much extra profit you gain per additional unit).
• P''(x) tells whether marginal profit is increasing or decreasing.

Concave down profit (P''(x)<0): marginal profit is decreasing. You may still be making more profit as you produce more, but each additional unit adds less profit than the previous one (diminishing returns).

Concave up profit (P''(x)>0): marginal profit is increasing. This can happen when scaling up reduces average costs or demand effects strengthen as production grows.

Context example: interpreting concavity without redoing earlier optimization steps

Suppose a company models profit by P(x)=100x-2x^2 (units in hundreds, profit in thousands, for example). Then P''(x)=-4<0 for all x, so profit is concave down everywhere: marginal profit decreases steadily as production increases. Even before finding any maximum, concavity alone tells you the “shape story”: increasing at first, but with diminishing returns, eventually turning downward.

Real Context: Population Growth That Slows Down

Let N(t) be a population over time.

• If N(t) is increasing and concave up (N''(t)>0), the growth rate N'(t) is increasing: growth is accelerating.
• If N(t) is increasing and concave down (N''(t)<0), the growth rate is decreasing: growth is slowing (still growing, but at a decreasing rate).

A common real pattern is: early concave up (rapid acceleration), then after resource limits appear, concave down (growth slows). The time when concavity changes corresponds to an inflection point: the moment the growth rate stops increasing and starts decreasing.

Mini-Lab 3: Describe a “Slowing Growth” Graph in Words

Setup: Draw an increasing curve that starts steepening (concave up) and later starts leveling off (concave down). Mark the transition point.

• Label one point in the early region and write: “At this time, the population is increasing and concave up, so the growth rate is ______.”
• Label one point in the later region and write: “At this time, the population is increasing and concave down, so the growth rate is ______.”
• Circle the transition and write: “This is an inflection point because ______.”

Common Pitfalls and How to Avoid Them

• Mixing up “increasing” with “concave up”: increasing means f'>0; concave up means f''>0. A function can be decreasing and concave up (going down but leveling off).
• Declaring inflection points from f''=0 alone: always check for a sign change in f''.
• Forgetting where f'' is undefined: those x-values can also be candidates (as long as the function exists there and concavity changes).
• Not using a sign chart: solving f''=0 is not enough; concavity is about intervals, so test intervals.