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Question

If f:[0,4π][0,π] be defined by f(x)=cos1(cos x)Then, the number of points xϵ[0,4π] satisfying the equation f(x)=10x10, is ___

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Solution

(i) Using definition of f(x)=cos1(x), we trace the curve f(x)=cos1(cos x).
(ii) The number of solutions of equations involving trigonometric and algebraic functions and involving both functions are found using graphs of the curves

We know that,cos1(cos x)=⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪x,If xϵ[0,π]2πx,If xϵ[π,2π]2π+x,If xϵ[2π,3π]4πx,If xϵ[3π,4π]

For above graph, it is clear that y=10x10 and y=cos1(cos x) intersect at three distinct points, so number of solutions is 3.


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