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Let f:[0,4pi]vec[0,pi] be defined by f(x...

Let `f:[0,4pi]vec[0,pi]` be defined by `f(x)=cos^(-1)(cosx)dot` The number of points `x in [0,4pi]s a t i sfy ingt h e` equation `f(x)=(10-x)/(10)` is____

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The correct Answer is:
3
(i) Using definition of ` f (x) = cos ^(-1)(x)`, we trace the curve ` f(x) = cos ^(-1) ( cos x ) `.
(ii) The number of solutions of equations involving tigonometric an algebric functions and involving both functions are found using graphs of the curves.
We know that`, cos ^(-1) ( cos x ) = {{:(x",",, if x in [0"," pi]), (2 pi - x ",",, if x in [ pi"," 2pi]), (- 2pi + x ",",, if in [ 2pi"," 3pi]), ( 4pi - x ",",, if x in [ 3pi "," 4pi]):}`

From above graph, it is clear that `y = (10 - x)/(10) and y = cos ^(-1) ( cos x)` intersect at three distinct points, so number of solutions is 3.
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