Question

How do you use the Intermediate Value Theorem to show that the polynomial function `2tanx - x - 1 = 0` has a zero in the interval [0, pi/4]?

Answer 1

`f(x)=2tan(x)-x-1` is continuous on the interval `[0,pi/4]`
and has a value `>0` and a value `<0`
therefore
by the Intermediate Value Theorem
it has ha value `=0`

Explanation:

`f(0) = 2tan(0)-0-1 = -1`

`f(pi/4)=2tan(pi/4)-pi/4-1`
`color(white)("XXX")=2(1)-pi/4-1`
`color(white)("XXX")=1-pi/4`
(and since `pi/4<1`)
`f(pi/4) > 0`