Question

true or false. for a continuous function, of f(2) is negative and f(3) is positive, then there must be a zero between x=2 and x=3​

Answer 1

It's true. In general, if a function is continuous, and you have

$f(x_1)=y_1,\quad f(x_2)=y_2$

then $f(x)$ spans all the possible values between $y_1$ and $y_2$ in the interval $[x_1,x_2]$

So, in particular, if you have

$f(2) = y_2 < 0,\quad f(3) = y_3 > 0$

you can conclude that in the interval $[2,3]$, $f(x)$ span all the values between $y_2$ and $y_3$, which include 0 because $y_2$ is negative and $y_3$ is negative.