The second derivative can be used to determine whether a critical point is a local maximum, a local minimum, or a saddle point. We call such a test second derivative test.

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A derivative of a derivative is known as second derivative. Second derivative is often used to measure curvature.

Second Derivative

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

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Second Derivative Test

A function $f(x)$ is considered convex if the line segment connecting any two points on its graph lies on or above the graph itself. From a calculus perspective, this geometric property corresponds to a non-negative second derivative ($f''(x) \ge 0$), which signifies that the function's slope, given by the first derivative ($f'(x)$), is non-decreasing.

Convex Function

The second derivative of a function in a specific direction, represented by a unit vector $$d$$, is given by $$d^T H d$$, where $$H$$ is the Hessian matrix of the function.

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