Question

Show by calculation that the equation cosx-2x=0 has a root between x=0 and x=0.785.how to show this ?

Answer 1

Evaluate `cosx-2x` at `x=0` and `x=0.785` and use the intermediate value theorem.

Explanation:

To show this, we will use the intermediate value theorem since `cosx - 2x` is a continuous function.

Let `f(x) = cosx - 2x`.

Let `color(red)(x=0)`.

`f(color(red)(0)) = cos(color(red)(0))-2(color(red)(0)) = 1`

Let `color(blue)(x=0.785)`.

`f(color(blue)(0.785)) = cos(color(blue)(0.785))-2(color(blue)(0.785)) ~~ -0.57`

But if `f(0) = 1` and `f(0.785) = -0.57`, at some point between, `f` must be equal to `0` since it is a continuous function.