Problems

Find the highest common factor of the two polynomials \begin{align*} f(x) &= x^4 - 13x^3 + 58x^2 - 96x + 36 \\ g(x) &= x^4 - 11x^3 + 36x^2 - 24x - 36. \end{align*} Find all the roots of the equation \(f^2(x) - g^2(x) = 0\).

Show that the real cubic equation \[ x^3+ax^2+b=0 \] has three real zeros if and only if \[ 27b^2+4a^3b \le 0. \]

Show that, if \(\alpha+\beta+\gamma = \pi\), \[ (\sin 2\alpha + \sin 2\beta + \sin 2\gamma)(\cot\alpha\cot\beta\cot\gamma - \cot\alpha - \cot\beta - \cot\gamma)+4=0. \]

Prove that \[ \lim_{n\to\infty} \left(\frac{\pi}{n}\right)^2 \sum_{\nu=0}^{n-1} (n-\nu)\sin\left(x+\frac{\pi\nu}{n}\right) = 2\sin x + \pi \cos x. \]

Find the number of stationary values of the function \(y=x^2+6\cos x\), distinguishing between maxima and minima, and find the number of points of inflexion.

1. If \(f_n(x) = \dfrac{d^n}{dx^n} \dfrac{\log x}{x}\) for \(x>0\), \(n=0, 1, 2, \dots\), show that \[ f_{n+1}(x) + (n+1)x^{-1}f_n(x) = (-1)^n n! x^{-n-2}. \]
2. If \(g_n(x) = \dfrac{d^n}{dx^n} \dfrac{3x^2+2x+1}{(x^2+1)(x+1)}\), show that for \(n=0, 1, 2, \dots\) \[ g_{4n+1}(1) = -4^{-(2n+1)}(4n+1)! \]

Evaluate \[ \int_0^\infty \frac{x^2\,dx}{(1+x^2)^{5/2}} \] and \[ \int_{-\infty}^\infty \frac{dx}{(e^{x/2}+1)(e^{-x/2}+1)}. \]

Find the volume of the body defined by \(z^2 \le e^{-(x^2+y^2)}\) and \(x^2+y^2 \le a^2\).

Prove that \(\displaystyle\int_0^x \frac{\sin y}{y}\,dy\) is positive when \(x\) is positive.

Show that, if the variables \(x, y\) and \(r, \theta\) are connected by the relations \[ x=r\cos\theta, \quad y=r\sin\theta \] and if \[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0, \] then \[ \frac{\partial^2 f}{\partial r^2} + \frac{1}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\frac{\partial^2 f}{\partial\theta^2} = 0. \]