#1. sorry, but if f(x) is continuous on a closed interval, it has to have a maximum and a minimum value. Now, if you mean no local max or min, then that just means that f'(x) is never zero in the interval. Also, by the intermediate value theorem, f(c) must be zero in the interval.
So. You know that f(x) is not continuous. Try to set up a step function, so it jumps from 1 to -1 in the interval.
#2. consider 10 tanh(x)
1. Construct a function f(x) that satisfies the following conditions:
I. Its domain is all real numbers.
II. It has no maximum and no minimum on the interval [ 1,3] .
III. It satisfies f(1) = 1 and f(3) = –1, but there does not exist a c between 1 and 3 such that f(x) = 0.
2. Construct a function f(x) that satisfies the following conditions:
I. f is continuous for all x.
II. –10< = f(x) < = 10 for all x.
III. f does not have a maximum or minimum on the interval [negative infinity, infinity].
1 answer
1 answer