Earlier, we noticed how the first derivative f^{\prime}(x) can be used to find maximum and minimum values.

Similarly, we can find maximum and minimum values for f^{\prime}(x) by using its derivative, f^{{\prime}{\prime}}(x) i.e. the second derivative.

Recall that f^{\prime}(x) is interpreted as the rate of increase/decrease/change, therefore f^{{\prime}{\prime}}(x) is capable of finding the maximum and minimum rate of increase or decrease.

A table is constructed below to illustrate the cases.

The first derivative of a function is known as f^{\prime}(x). Similarly, we denote the second derivative of a function as f^{{\prime}{\prime}}(x) or \frac{d^2y}{dx^2}.

If f^{{\prime}{\prime}}(x)>0 for an interval, the gradient of f(x) is increasing in the interval. The curve is said to be concave up (or have a ‘smiley face’)