# How am I supposed to compile a proof to show that a function has only $1$ real zero. [closed]

For something like $$x^7 + 2x^3 + 3x - 7$$

Would I apply Rolle's theorem ?

• What does this question have to do with group theory? – 5xum Mar 3 '16 at 10:50
• the example function you provided is monotonic increasing... – Harry Mar 3 '16 at 10:55
• Yes, Rolle's theorem helps a lot. – bartgol Mar 3 '16 at 21:26

$$7x^6 + 6x^2 + 3$$
Since $x^7, x^3$ and $x$ are all strictly increasing, their sum is strictly increasing. The "$-7$" is just a translation. It follows the function is strictly increasing, therefore injective. In particular, there can be only one zero.
Observing that $$\lim_{x\to -\infty} f(x) = -\infty$$ and $$\lim_{x\to+ \infty} f(x) = +\infty$$ and that $f$ is continuous, there is at least one so you're done.