If is continuous on [4,7] which of the following statement must be true?
I has a maximum value on [4,7]
II has a minimum value on [4,7]
III f(7)>f(4) .
IV Lim as x approaches 0 f(x)=f(6)
Given that `f(x)` is continuous on [4,7]:
(1),(2) `f(x)` has a maximum and a minimum on [4,7] is true. The Extreme Value Theorem state that if f is a continuous function on a closed interval then it achieves both its maximum and minimum at least once in the interval.
(3) `f(7)>f(4) ` is false. We are not told that `f` is an increasing function.
(4) `lim_(x->0)f(x)=f(6)` is false. If you meant `lim_(x->6)f(x)=f(6)` then this is true -- a function is continuous at a point `c` if `lim_(x->c)f(x)` exists, `f(c)` exists, and `lim_(x->c)f(x)=f(c)` .
If a function is continuous on an interval, then a particular value is assigned to the function for each x value within the interval. So both I and II are true.
Incidentally, this is not the case if you just say "f is continuous," which implies "f is continuous on its domain." For example, f(x) = 1/x is continuous, but does not have a maximum or minimum value.