Science:Math Exam Resources/Courses/MATH104/December 2013/Question 01 (j)/Solution 1

The correct answer is (E).

To show that E is correct, we can use the intermediate value theorem twice. In order for f to be a continuous function that connects $\displaystyle (-1,4)$ and $\displaystyle (3,-2)$, we know that it must have some value at $\displaystyle x=0$ by the fact that the function is continuous for all real values of x. Thus, it crosses the y axis. As these two points live both above and below the x axis, we see that $\displaystyle f(-1)\geq 0\geq f(3)$ and since f is continuous, the intermediate value theorem states that there is some value c in the interval $\displaystyle (-1,3)$ such that $\displaystyle f(c)=0$. Thus the answer is (E).

Note. For the curious minded, the graph below gives a counter example to all of the other 4 conditions.