How do I use the intermediate value theorem to determine whether #x^5 + 3x^2 - 1 = 0# has a solution over the interval #[0, 3]#?
You need only use Bolzano's Theorem. See below
Bolzano's Theorem state that "If f continous over [a,b] and f(a)·f(b)<0, then there is c in (a,b) such that f(c)=0"
Note that f(a)f(b)<0 means that f(a) and f(b) have diferent sign in a and b
In our case
By Bolzano's theorem there is a value c between 0 and 3 such that f(c)=0
In this case (and in general) you can do a fine tunnig of c. For example
Then that c is between 0 and 1
We can`t assume that there is no other c between 1 and 3 because f has no sign change there...
Please see the explanation below.
The intermediate value theorem states :
If the function
is a continouos function on the interval
This is the application of the intermediate value theorem.