Show that the equation $2x^4-9x^2+4 = 0$ has at least one solution in $(0,1)$.
It is not possible to show it by Bolzano's theorem because neither 0 nor 1 are in the given interval, is it? Is there any way to do it other than solving it agebraically or analysing its graph? By the way, the root in $(0, 1)$ is $x = \frac{1}{\sqrt2}$.
Thanks in advance.
Best Answer
What about the fact that $f(0) > 0$, and $f(1) < 0$, so because the function is continuous, the function must take on all y-values in between $f(0)$ and $f(1)$. This includes $0$.