Maxima and Minima of Functions of One Variable A maximum of a function is a point where the function reaches its highest value. A minimum is a point...
A maximum of a function is a point where the function reaches its highest value. A minimum is a point where the function reaches its lowest value.
The values of a function at its maximum and minimum points can be found by finding the highest and lowest values of the function's derivative. The critical points are the points where the derivative is equal to zero, since these points indicate where the function's slope is zero and therefore, the function's rate of change is zero.
The first derivative tells us the direction of the function's increase or decrease. A positive first derivative indicates increasing function, while a negative derivative indicates decreasing function.
The second derivative gives us the concavity of the function. A positive second derivative indicates concave up, while a negative second derivative indicates concave down.
By understanding the relationship between the first and second derivatives, we can identify the critical points and determine if they correspond to maximums or minima.
Examples:
At x = 0, the function reaches its minimum value of 0.
At x = 0 and x = 3, the function takes the value 0, which is the critical point.
At x = π/2, the function reaches its maximum value of 1, while at x = 0, it reaches its minimum value of -1