The intermediate value theorem says that if the signs of f(a) and f(b) differ, there is a root of f between a and b.

You have the expression

tan(x) = 2x

This can be formulated as a function f(x) whose root we are seeking

It is proposed that there is a root between x=0 and x=1.4. We can check the signs.

F(0) = 0 (

f(1.4) = tan(1.4) - 2*1.4 = 5.7979 - 2.8000 = 2.9979

This check is inconclusive, as the given interval is an open interval that does not include x=0.

We can also check x=.7 (arbitrarily, the midpoint of the interval)

f(.7) = tan(.7) - 2*.7 = .8423 - 1.4000 = -.5577

There is a change of sign in f(x) between x=0.7 and x=1.4. The intermediate value theorem tells us there is a root in the range (0.7, 1.4), a range that is included in the given range.

You have the expression

tan(x) = 2x

This can be formulated as a function f(x) whose root we are seeking

**.**f(x) = tan(x) - 2xIt is proposed that there is a root between x=0 and x=1.4. We can check the signs.

F(0) = 0 (

**this is a root**)f(1.4) = tan(1.4) - 2*1.4 = 5.7979 - 2.8000 = 2.9979

This check is inconclusive, as the given interval is an open interval that does not include x=0.

We can also check x=.7 (arbitrarily, the midpoint of the interval)

f(.7) = tan(.7) - 2*.7 = .8423 - 1.4000 = -.5577

There is a change of sign in f(x) between x=0.7 and x=1.4. The intermediate value theorem tells us there is a root in the range (0.7, 1.4), a range that is included in the given range.