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What is the minimum angle by which the coordinate axes have to be rotated in anticlockwise sense (in Degrees), such that the function f(x) = 3x3 + 5x + 1016 has at least one Rolles point

What is the minimum angle by which the coordinate axes have to be rotated in anticlockwise sense (in Degrees), such that the function f(x) = 3x3 + 5x + 1016 has at least one Rolles point Correct Answer 180⁄π tan-1(5)

For the transformed function to have a Rolles point is equivalent to the existing function having a Lagrange point somewhere in the real number domain, we are finding the point in the domain of the original function where we have f'(x) = tan(α) Let the angle to be rotated be α We have f'(x) = 9x2 + 5 = tan(α) 9x2 = tan(α) – 5 For the given function to have a Lagrange point we must have the right hand side be greater than zero, so tan(α) – 5 > 0 tan(α) > 5 α > tan-1(5) In degrees we must have, αdeg > 180⁄π tan-1(5).

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