SOLUTION: suppose f(x) is a continuous, monotone function on the interval [0,2]. explain why the maximum of f(x) is either f(0) or f(2).

Algebra ->  Functions -> SOLUTION: suppose f(x) is a continuous, monotone function on the interval [0,2]. explain why the maximum of f(x) is either f(0) or f(2).       Log On


   

Question 546588: suppose f(x) is a continuous, monotone function on the interval [0,2]. explain why the maximum of f(x) is either f(0) or f(2).
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
By the extreme value theorem, f has a maximum. If the x-value corresponding to the maximum occurred in the interval (0,2), then the sign of f'(x) would have to change (e.g. if the function is differentiable, then f'(x) = 0 for some x strictly between 0 and 2). This cannot happen, so the maximum of f is either at x = 0 or x = 2.