Concavity Calculator
Find where a graph is concave up or concave down, detect candidate inflection points, and read the sign of the second derivative with a function or sign-table workflow.
Why This Concavity Calculator Is Useful
Function mode
Enter a common calculus expression like x^3 - 3x or sin(x) and get interval-by-interval concavity analysis.
Sign-table mode
If you already know the second-derivative signs from a graph or homework table, you can classify concavity instantly without re-deriving the function.
Inflection clues
The page points out candidate inflection points where the second derivative changes sign or touches zero inside the tested range.
Concavity Rules
Concave up
If f''(x) is positive, the graph bends upward like a cup. Slopes increase as x increases.
Concave down
If f''(x) is negative, the graph bends downward like a cap. Slopes decrease as x increases.
Inflection point
A candidate inflection point is where concavity changes sign. That is the main place to check on a graph or derivative test.
Test-point method
Pick a point in each interval and evaluate the sign of f''(x). This tool automates the same logic when you use the sign-table mode.
Frequently Asked Questions
How do I know if a function is concave up or concave down?
If the second derivative is positive on an interval, the graph is concave up. If the second derivative is negative, the graph is concave down. This calculator checks the sign of the second derivative across the interval and classifies each section for you.
What is an inflection point?
An inflection point is where the graph changes concavity. In calculus terms, that usually happens when the second derivative changes sign from positive to negative or from negative to positive.
Do I need the full derivative to use this page?
No. In function mode, you can enter the original function and the calculator estimates concavity numerically from finite differences. In sign-table mode, you can enter the interval signs directly if you already know them from homework or a graph.
What does the sign-table mode do?
It lets you enter boundary x-values and the second-derivative sign for each interval. The tool then labels each interval as concave up or concave down and marks every sign change as a candidate inflection point.
Is this a symbolic calculus solver?
No. This page is designed as a practical concavity analyzer. It is fast and useful for common classroom functions, but it does not replace a full symbolic CAS or proof-based derivative solver.
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