Question

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) = 3x^3 + 5x + 1016 has at least one Rolles point

(a) ^π⁄180 tan^-1(5)

(b) tan^-1(5)

(c) ^180⁄π tan^-1(5)

(d) -tan^-1(5)

The question was asked by my college director while I was bunking the class.

My enquiry is from Lagrange’s Mean Value Theorem topic in chapter Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (c) ^180⁄π tan^-1(5)

The explanation is: 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) = 9x^2 + 5 = tan(α)

9x^2 = 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,

α^deg > ^180⁄π tan^-1(5).