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

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Answer 1 · 2018-07-27T02:18:39

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

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.