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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​

Respuesta :

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

[tex] f(x_1)=y_1,\quad f(x_2)=y_2[/tex]

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

So, in particular, if you have

[tex] f(2) = y_2 < 0,\quad f(3) = y_3 > 0[/tex]

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

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