The second derivative is a measure of how the rate of change of a function is itself changing, obtained by differentiating a function twice. There are two common notations for the second derivative:
For example, if we have a function $f(x) = x^3$, its first derivative is $f'(x) = 3x^2$, and its second derivative is $f''(x) = 6x$.
The second derivative provides crucial information about the shape and behavior of a function's graph:
Consider the function $f(x) = x^3$:
The function is concave up when $x > 0$ and concave down when $x < 0$.
< 0$. The inflection point occurs at $x = 0$.
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