Math calculator
Concavity Calculator
Use the second derivative sign to identify concave-up and concave-down regions. The page traces how the inputs become concavity intervals before rounding.
Concavity inputs
Enter the source values
A practical accuracy check
This Concavity calculation is based on Function f(x), Interval start, and Interval end. Use scale as a reasonableness test. Estimate the likely sign and order of magnitude from the graph or dominant term, then compare that expectation with the computed value before copying it into later work.
For this concavity result, if the result will feed another calculation, save several unrounded digits and the settings that produced them. Round only when presenting the final quantity, after checking that greater resolution does not move it materially.
Testing Concavity beyond the example
Inspect the domain of Function f(x) before using Concavity. Keep exact Concavity work separate from Interval start.
Test Concavity on a constant or linear function. Refine any numerical Concavity step and compare the approximation.
A practical Concavity check is to simplify Function f(x) and leave Interval start unchanged. The resulting Concavity intervals should be easy to estimate, giving a reference point for the less convenient values in the original problem.
Cross-checking Concavity
Work backward from the displayed Concavity intervals once. Ask whether Function f(x) can produce that Concavity intervals under Interval start. If the reverse Concavity relationship fails, recheck the entry order and any convention attached to Interval end.
Reviewing Concavity in context
Concavity is determined by the sign of the second derivative over the interval. For that connected step, see second-derivative test.
Change only Function f(x) during a Concavity trial. With Interval start unchanged, the movement in Concavity intervals should follow the stated mathematical relationship.
Record the first Concavity setup before editing Function f(x). A later value of Concavity intervals should remain attached to its own inputs.
Keep Function f(x), Interval start, and the unrounded Concavity intervals together. Those values let a later reader reproduce this particular Concavity case.
Delay rounding Concavity intervals until the next Concavity step is known. Extra digits can prevent a display-rounding difference from accumulating.
Questions about Concavity
What does f″>0 mean?
Concave up.
Does f″=0 prove inflection?
No.
Are intervals bounded by the input?
Yes.