The graph of y = f (x) is concave upward on those intervals where
y = f "(x) > 0.
The graph of y = f (x) is concave downward on those intervals where
y = f "(x) < 0.
What is concave upward and concave downward?
If the curve "bends" upward, a portion of the graph of f has a concave upward slope. For instance, the entire popular parabola y=x^2 is concave upward. If the curve "bends" downward, a portion of the graph of f is concave.
Finding domain values where f′′(x) = 0 or where f′′(x) does not exist is the first step in identifying intervals where a function is concave upwardly or downwardly.
The second derivative of the function should then be tested for all intervals surrounding these values.
If the sign of f′′(x) changes, then the point (x, f(x)) is where the function inclines.
Hence, the graph of y = f (x) is concave upward on those intervals where
y = f "(x) > 0.
The graph of y = f (x) is concave downward on those intervals where
y = f "(x) < 0.
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