Sketch the graphs of the curves y=x and y=cos x. Use Intermediate Value Theorem to identify an interval where cos x = x has a solution.

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The graph of y = x and y = cos x is shown below. (y = x is in black and y = cos x is in red)

The solution of cos x = x has to be determined.

cos x = x

=> cos x - x = 0

...

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The graph of y = x and y = cos x is shown below. (y = x is in black and y = cos x is in red)

The solution of cos x = x has to be determined.

cos x = x

=> cos x - x = 0

For x = 0.7

cos x - x = 0.0648

for x = 0.8

cos x - x = -0.1032

By the intermediate value theorem a solution of cos x = x lies in the interval (0.7, 0.8)

This can be seen when f(x) = cos x - x is plotted.

It is seen that the graph crosses the x-axis at a value of x lying between 0.7 and 0.8.

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