Question

Let f be a continuous function on the closed interval [ − 2 , 5 ]. If f((-2)=-7 and f(5)=1, then the Intermediate Value Theorem guarantees that

Answer 1

Answer:

The function has at least 1 zero within the interval [-2,5].

Step-by-step explanation:

The intermediate value theorem states that, for a function continuous in a certain interval $[a,b]$, then the function takes any value between $f(a)$ and $f(b)$ at some point within that interval.

This theorem has an important consequence:

If a function $f(x)$ is continuous in an interval [a,b], and the sign of the function changes at the extreme points of the interval:

$f(a)>0\\f(b)$ (or viceversa)

Then the function f(x) has at least one zero within the interval [a,b].

We can apply the theorem to this case. In fact, here we have a function f(x) continuous within the interval

[-2,5]

And we also know that the function changes sign at the extreme points of the interval:

$f(-2)=-70$

Therefore, the function has at least 1 zero within the interval [-2,5], so there is at least one point x' within this interval such that

$f(x')=0$