# How do you show that #1 + 2x +x^3 + 4x^5 = 0# has exactly one real root?

##### 2 Answers

See explanation.

#### Explanation:

First if we look at the limits of our function we see, that:

To show, that the function has **only one** real root we have to show, that it is monotonic in the whole

Now we have to solve:

To do this we substitute

This experssion is positive for all

So we can conclude, that it has only one real root.

QED

See the explanation section.

#### Explanation:

Let

** #f# has at least one real zero** (and the equation has at least one real root).

This

Suppose that

(

and

However,

(Look at each term. Both

That means that no

Conclusion

If there were two (or more) zeros, then we could make

But, clearly, we cannot make