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.