f(x) has an inflection point at (c, f(c)) when
the concavity changes at f”(c) = 0 or f”(c) is undefined
f”(c) changes sign at point c. In other words, f”(c) is zero at point c and changes from positive to negative or negative to positive at that point.
An inflection point is a point on the graph of a function at which the concavity changes. At an inflection point, the second derivative of the function changes sign. If the second derivative changes from positive to negative at the inflection point, the graph is said to be concave down; if it changes from negative to positive, the graph is said to be concave up.
In simple terms, the second derivative tells us whether the graph of a function is concave up or concave down. If the second derivative is positive, the graph is concave up, and if it’s negative, the graph is concave down. At an inflection point, the second derivative changes sign, indicating a change in concavity.
Therefore, if f”(c) changes sign from positive to negative or negative to positive at point c, then f(x) has an inflection point at that point.
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