Final answer:
To find the local and global extrema of the function f(x) = x(25 - x), we first find the derivative and set it equal to zero. Next, we evaluate the function at the critical point and the endpoints of the interval [0, 20]. The local maximum is 156.25 and the global maximum is also 156.25.
Step-by-step explanation:
To find the local and global extrema of the function f(x) = x(25 - x), we can start by finding the critical points. Critical points occur where the derivative of the function is equal to zero or undefined. Let's find the derivative of f(x) first:
f'(x) = 25 - 2x
Setting f'(x) equal to zero, we get:
25 - 2x = 0
Solving for x:
x = 12.5
The critical point is x = 12.5. Now, let's evaluate f(x) at the endpoints of the interval [0, 20] and the critical point:
f(0) = 0
f(20) = 0
f(12.5) = 12.5(25 - 12.5) = 156.25
Therefore, the local maximum is f(12.5) = 156.25 and the global maximum value on the given interval is also f(12.5) = 156.25.