Final answer:
To find the maximum and minimum of a function, you can follow these steps: 1. Take the derivative of the function to find the critical points. 2. Evaluate the function at these critical points and also at the endpoints of the given interval. 3. The maximum and minimum values of the function will be the highest and lowest y-values obtained from the previous steps.
Step-by-step explanation:
To find the maximum and minimum of a function, you can follow these steps:
- First, take the derivative of the function to find the critical points. The critical points are the x-values where the derivative equals zero or is undefined.
- Evaluate the function at these critical points and also at the endpoints of the given interval to find the corresponding y-values.
- The maximum and minimum values of the function will be the highest and lowest y-values obtained from the previous step, respectively.
For example, if you have the function f(x) = x^2 - 4x + 3 on the interval 0 ≤ x ≤ 5, you can find the maximum and minimum as follows:
- Take the derivative of f(x) to get f'(x) = 2x - 4. Set f'(x) = 0 and solve for x to find the critical point x = 2.
- Evaluate f(x) at x = 2, as well as at the endpoints x = 0 and x = 5. The corresponding y-values are f(2) = -1, f(0) = 3, and f(5) = 8.
- The maximum value of the function is 8, obtained at x = 5, and the minimum value is -1, obtained at x = 2.