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# Second Derivatives

## 5.7 Maximum Rates of Increase or Decrease

#### Maximum Rates of Increase or Decrease

• 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.
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## 5.5 Second Derivatives and Concavity

#### Second Derivatives and Point of Inflection

• 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’)
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