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Can Rolle's theorem be applied to each of the following functions? Find 'c' in case it can be applied.
i. f(x) = sin2 x on the interval [0, π].
ii. f (x) = x² + 4 on [-2, 2].
iii. f(x) = sinx + cos x on
iv. f(x) = x³- 2x on [0, 1].
Reduce the equation
xy" + y' + xy = 0, x > 0
into Bessel's equation and hence write its general solution in terms of Bessel's functions.
Determine the solution of the undamped (forced vibrations) system mü + Ku = Fo cos wt, u(0) = 0, u(0) = 1 When, w ≠
Find 'c' of Cauchy's mean value theorem for the function f(x) = 2.ln(x) and g(x) = x²-1 in the interval [2,3].
Use the method of Laplace transforms to find the solution of the initial value problem y" + 9y = 6 cos 3t, y(0) = 2, y'(0) = 0
Find the height of a right cylinder with greatest lateral surface area that may be inscribed in a given sphere of radius R.