Let f (a function on an interval)

Question

Let f be a function such that f(4) = –1 and f(6) = 1. Which of the following statements must be true?

I. f has a root on the interval [4, 6].

II. f has an absolute maximum on the interval [4, 6].

none

I only

II only

I and II

Answer 1

I would be True, but only if f(x) is a continuous function

f(x) must cross the x axis somewhere bewteen x=4 and x=6, if the function is continuous. That intersection point is the root or zero.

But for a discontinuous function, there may be no root in the given interval. For example, f(x) = 1/(x-5)

has f(4) = 1/(4-5) = 1/-1 = -1

and f(6) = 1/(6-5) = 1/1 = 1

while the graph is discontinuous at x=5

as f(5) = 1/(5-5) = 1/0 which is undefined

the graph never crosses the x axis

The answer is "none" neither I nor II

II is also false

II would be true if the function were continuous, then the absolute maximum is either a relative maximum, if any, or f(6)= 1, given that the interval includes the endpoints

if the question had included any information indicating the function was continuous, then the answer would be I and II are both true.