# 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

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 f(0)<0 f(1)=3>0

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...

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Please see the explanation below.

The intermediate value theorem states :

Here,

This is the application of the intermediate value theorem.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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