Concavity & Inflection Points Cheat Sheet

A printable reference covering concavity, second derivatives, inflection points, sign charts, and the second derivative test for grades 11-12.

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Concavity describes how the graph of a function bends, which helps students understand the shape of curves beyond increasing and decreasing behavior. This cheat sheet explains how to use the second derivative to identify concave up and concave down intervals. It also shows how to find and confirm inflection points. These ideas are essential for curve sketching, optimization, and interpreting motion or rate-of-change graphs. The core rule is that $f''(x) > 0$ means the graph is concave up, while $f''(x) < 0$ means the graph is concave down. Possible inflection points occur where $f''(x) = 0$ or where $f''(x)$ is undefined. A point is an inflection point only if concavity changes across that $x$ -value. The second derivative test also uses $f''(c)$ to classify certain critical points as local maxima or local minima.

Key Facts

A function is concave up on an interval when $f''(x) > 0$ on that interval.
A function is concave down on an interval when $f''(x) < 0$ on that interval.
A possible inflection point occurs where $f''(x) = 0$ or where $f''(x)$ does not exist.
An inflection point exists at $x = c$ only if $f''(x)$ changes sign as $x$ passes through $c$ .
If $f''(c) > 0$ and $f'(c) = 0$ , then $f(c)$ is a local minimum by the second derivative test.
If $f''(c) < 0$ and $f'(c) = 0$ , then $f(c)$ is a local maximum by the second derivative test.
If $f''(c) = 0$ , the second derivative test is inconclusive and another method must be used.
For position $s(t)$ , acceleration is the second derivative $s''(t)$ , so concavity describes whether velocity is increasing or decreasing.

Vocabulary

Concavity: Concavity describes whether a graph bends upward or downward over an interval.
Concave Up: A graph is concave up where $f''(x) > 0$ , meaning its slopes are increasing.
Concave Down: A graph is concave down where $f''(x) < 0$ , meaning its slopes are decreasing.
Inflection Point: An inflection point is a point on a graph where concavity changes from up to down or from down to up.
Second Derivative: The second derivative $f''(x)$ measures the rate of change of the first derivative $f'(x)$ .
Second Derivative Test: The second derivative test uses the sign of $f''(c)$ at a critical point to classify a local maximum or minimum.

Common Mistakes to Avoid

Calling every solution of $f''(x) = 0$ an inflection point is wrong because concavity must actually change sign.
Forgetting to check where $f''(x)$ is undefined is wrong because inflection points can occur at undefined second derivative values if the function is still continuous there.
Using $f'(x)$ instead of $f''(x)$ to determine concavity is wrong because the first derivative tells increasing or decreasing, not bending direction.
Applying the second derivative test when $f'(c) \neq 0$ is wrong because the test classifies only critical points where $f'(c) = 0$ or $f'(c)$ is undefined.
Assuming $f''(c) = 0$ means neither maximum nor minimum is wrong because the second derivative test is inconclusive, so another method is needed.

Practice Questions

1 For $f(x) = x^3 - 6x^2 + 4$ , find $f''(x)$ and determine the intervals where the graph is concave up and concave down.
2 Find all inflection points of $g(x) = x^4 - 4x^3$ by solving for possible values and checking for a sign change in $g''(x)$ .
3 Use the second derivative test to classify the critical point of $h(x) = x^2 - 8x + 3$ .
4 Explain why a point where $f''(x) = 0$ is not automatically an inflection point.

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