Some authors use concave for concave down and convex for concave up ... In other words, the point 0 is a point of inflection even though f ''(0) does not exist.
If f′′(x)=0 and the concavity of the graph changes (from up to down or vice ... value of x between each inflection point; the concavity will not change without an ...
When the function y = f (x) is concave down, the graph of its derivative y = f '(x) is ... At all points h '(x) = 0 and h "(x) = 0 but there is no point of inflection.
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Similarly, if f ''(x) < 0 on (a,b), then the graph is concave down. Conversely ... is monotonic. Hence its derivative, i.e., the second derivative, does not change sign.
The same goes for f(x) concave down, but then f ''(x) is non-positive. Saying f ''(x) ≠ 0 is not enough to determine the concavity of f(x), because f ...
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