maximum value of a function
when a parabola opens downward, the y-value of the vertex
The maximum value of a function is the highest value that the function can achieve within its domain. To find the maximum value of a function, you need to follow these steps:
1. Find the critical points: These are the points at which the derivative of the function equals zero or is undefined.
2. Check the endpoints: If the function is defined over a closed interval, you must also check the function value at the endpoints of that interval.
3. Evaluate the function at the critical points and endpoints: Once you have identified all critical points and endpoints, you can plug these values into the function and evaluate it. The highest value among these will be the maximum or the global maximum if it is strictly greater than any other value of the function.
4. Determine if the maximum value is a maximum or a minimum: Depending on the function and interval, the highest value may be a maximum or a minimum. You can determine this by checking the sign of the second derivative of the function.
It is important to note that not all functions have a maximum value. For example, some functions, such as y = x, continue to increase indefinitely without ever reaching a maximum point.
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