Eq 12xsinx Use the intermediate value theorem to prove that

Eq: 1-2x=sinx Use the intermediate value theorem to prove that this equation has a root in the interval (0, pi/4) check all the criterions of IVT.

Solution

f(x) = 1-2x-sin(x)

f(0) = 1 > 0

f(/4) = 1 - /2 - 2/2 = -1.28 < 0

Thus by intermediate value theorem, there exists a c in (0 , /4), such that:

f(c) = 0

1-2c-sin(x) = 0

1-2c = sin(c)

so the equation a-2x = sin(x) has a solution in (0, /4)