Maxima and Minima from Calculus

One of the great powers of calculus is in the determination of the maximum or minimum value of a function. Take f(x) to be a function of x. Then the value of x for which the derivative of f(x) with respect to x is equal to zero corresponds to a maximum, a minimum or an inflexion point of the function f(x).

微积分的一大威力在于确定函数的最大或最小值。设f(x)为x的函数。则当f(x)对x的导数等于零时，对应的x值对应于函数f(x)的最大值、最小值或拐点。

For example, the height of a projectile that is fired straight up is given by the motion equations:

例如，竖直上抛的弹体高度由运动方程给出：

取 y₀ = 0，下面展示了高度 y(t) 的图像。

The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function y(t) plotted as a function of t. The derivative is positive when a function is increasing toward a maximum, zero (horizontal) at the maximum, and negative just after the maximum. The second derivative is the rate of change of the derivative, and it is negative for the process described above since the first derivative (slope) is always getting smaller. The second derivative is always negative for a "hump" in the function, corresponding to a maximum.

函数的导数可以几何上解释为数学函数 y(t) 随 t 变化的曲线的斜率。当函数向最大值增加时，导数为正，达到最大值时为零（水平），最大值之后导数为负。二阶导数是导数的变化率，如上所述过程中的二阶导数为负，因为一阶导数（斜率）始终在减小。对于函数中的“波峰”，二阶导数始终为负，对应于最大值。

For the simple function used in the example, there is only one maximum. More complex functions can have multiple maxima and minima, and the evaluation of the second derivative provides a way to distinguish them.

在示例中使用的简单函数只有一个最大值。更复杂的函数可能有多个最大值和最小值，而二阶导数的评估可以用来区分它们。