Prove that the geometric mean of a finite set of positive numbers is not greater than their arithmetic mean. When does equality occur? Find the volume of the greatest box whose sides of length \(x, y\) and \(z\) satisfy \[ 36x^2+9y^2+4z^2=36. \]
The three numbers \(X, Y\) and \(Z\) are related to the three numbers \(x, y\) and \(z\) by the two equations \[ \frac{X}{x+3y-z} = \frac{Y}{3x+4y-2z} = \frac{Z}{-x-2y+2z}. \] Find one set of constants \(\alpha, \beta, \gamma\) and \(\lambda\) so that each of these three ratios is equal to \[ \lambda \left(\frac{\alpha X + \beta Y + \gamma Z}{\alpha x + \beta y + \gamma z}\right) \] for all \(x, y\) and \(z\).
The numbers \(a_1, b_1, a_2, b_2, \dots\) and the numbers \(c_1, c_2, c_3, \dots\) are all positive and \[ 0 < h < \frac{a_1}{b_1} < \frac{a_2}{b_2} < \dots < \frac{a_n}{b_n} < \dots < H. \] Show that \[ h < \left( \frac{a_1^m c_1 + a_2^m c_2 + \dots + a_n^m c_n}{b_1^m c_1 + b_2^m c_2 + \dots + b_n^m c_n} \right)^{\frac{1}{m}} < H \quad (m=1, 2, 3, \dots). \] Show also that \[ h < \left( \frac{\frac{1}{b_1^m c_1} + \frac{1}{b_2^m c_2} + \dots + \frac{1}{b_n^m c_n}}{\frac{1}{a_1^m c_1} + \frac{1}{a_2^m c_2} + \dots + \frac{1}{a_n^m c_n}} \right)^{\frac{1}{m}} < H \quad (m=1, 2, 3, \dots). \]
A sequence of functions \(f_n(x)\), \(n=0, 1, 2, \dots\), is defined by \[ \begin{cases} f_0(x) = 1 \\ f_{n+1}(x) = (1+x)^{f_n(x)} \end{cases} \] Assuming that \(f_n(x)\) can be expanded in powers of \(x\), show that \[ f_n(x) = 1+x+x^2+\frac{3}{2}x^3+\dots \text{ for } n\ge3. \] Show that \(f_{n+1}(x)=f_n(x)+x^{n+1}+\text{higher powers of } x\). Deduce that the coefficient of \(x^m\) in the expansion of \(f_n(x)\) in powers of \(x\) is independent of \(n\) for \(n \ge m\).
Sketch the graph of a function \(f(x)\) that satisfies the conditions (i) \(f(0)=0\), (ii) \(f'(0)<0\), (iii) \(f''(x)>0\) for \(x>0\), (iv) \(f(x)\) tends to a limit as \(x\to\infty\). Also sketch the graph of a function \(g(x)\) that satisfies the conditions (i) \(g(0)=0\), (ii) \(g'(0)<0\), (iii) \(g''(0)>0\), (iv) \(\dfrac{g(x)}{x} \to 1\) as \(x\to\infty\).
Show that, if \(u=x^2\), \[ x\frac{d^2f(x)}{dx^2} - \frac{df(x)}{dx} = 4x^3 \frac{d^2f(x)}{du^2}. \] Find a function satisfying the equation \[ x\frac{d^2f(x)}{dx^2} - \frac{df(x)}{dx} - 4x^3 f(x) = 0 \] and containing two arbitrary constants.
Sketch the curve \[ y^2(1+x^2) = (1-x^2)^2, \] and find the area of its loop.
Evaluate \[ \int_0^{2\pi} \frac{\sin^2\theta d\theta}{2-\cos\theta}, \quad \int_{1}^2 \sqrt{\frac{x-1}{x+1}} \frac{dx}{x}. \]
If \[ I_n = \int_0^\infty \frac{dx}{(x+1)(x^2+1)^n}, \] show that \[ (2n+1)I_n - 2(3n+2)I_{n+1} + 4(n+1)I_{n+2} = \frac{1}{2n}. \]
Show that the arithmetic mean \(A=(a_1+\dots+a_n)/n\) of \(n\) positive numbers \(a_1, \dots, a_n\) is never less than the geometric mean \(G=(a_1 a_2 \dots a_n)^{1/n}\). If, further, \(a_i \ge 1\) for \(1 \le i \le n\) show that \[ G \ge (nA-n+1)^{1/n}. \]