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

1 Answer
Jul 27, 2018

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.