How to show there exist a solution?

Question

Show that there exists a real number between $0≤x≤2$ so that $$x^7+8x^2−10=0$$

I know that a solution exists and its approx 1.04 but how do I show it?

Answer 1

You can prove like that:

Since $f(x)$ is continuous and $f(0)\cdot f(2) < 0$, then the Bolzano's Theorem (Intermediate Value Theorem) confirms that there is a root in this interval.

Answer 2

ohh I get it now: I use this part of the intermediatiate value theorem $$x < y, f(x) = a ≤ c ≤ b = f(y) $$

Then there exists $$x≤s≤y$$ such that $$f(s)=C$$