Find necessary conditions to be satisfied by the coefficients \(a, b, c\) in order that \(ax^2 + 2bx + c\) may be positive for all real values of \(x\). Prove that these conditions are sufficient. Assuming that these conditions are satisfied, find, in terms of \(a, b, c, k\), the greatest value that \(h\) can have if \[ ax^2 + 2bx + c \ge h(x-k)^2 \] for all real values of \(x\).
Determine all sets of solutions \((x, y, z)\) of the equations \begin{align*} x + y + z &= a+b+c, \\ a^2x + b^2y + c^2z &= a^3+b^3+c^3, \\ a^3x + b^3y + c^3z &= a^4+b^4+c^4, \end{align*} where \(a, b, c\) are unequal, distinguishing the cases in which \(bc+ca+ab\) is different from or equal to zero.
Resolve \(x^{2n}+1\) into real quadratic factors, where \(n\) is a positive integer. Express \[ \frac{1}{x^{2n}+1} \] in partial fractions with these factors as denominators.
Find for what ranges of values of \(\theta\) between \(0\) and \(\pi\) each of the following inequalities is satisfied:
If \(y = a + x \log y\), where \(x\) is small, prove that \(y\) is approximately equal to \[ a + x\log a + \frac{x^2}{a}\log a \] and obtain the term in \(x^3\) in the expansion.
Find the limits, as \(x\) tends to \(\frac{1}{2}\), of the following expressions:
Prove that, if \(x\) is positive, \[ \frac{2x}{2+x} < \log(1+x) < x. \] Prove also that, if \(a\) and \(h\) are positive, \[ \log(a+\theta h) - \log a - \theta\{\log(a+h) - \log a\}, \] considered as a function of \(\theta\), has a maximum for a value of \(\theta\) between \(0\) and \(\frac{1}{2}\).
Prove that, if \(r_1, r_2\) denote the distances from two fixed points \(O_1, O_2\), of a variable point \(P\) in a fixed plane through \(O_1O_2\), and \(\theta_1, \theta_2\) the angles \(PO_1O_2, \pi - PO_2O_1\), then curves of the two families \begin{align*} r_1^m r_2^n &= \lambda, \\ m\theta_1 + n\theta_2 &= \mu, \end{align*} where \(m\) and \(n\) are constants and \(\lambda, \mu\) are variable parameters, cut orthogonally.
A function \(f(x)\) is defined, for \(x \ge 0\), by \[ f(x) = \int_{-1}^1 \frac{dt}{\sqrt{\{1-2xt+x^2\}}}. \] Prove that, if \(0 \le x \le 1\), \(f(x)=2\). What is the value of \(f(x)\) if \(x > 1\)? Has the function \(f(x)\) a differential coefficient for \(x=1\)?
Find the asymptotes of the curve \[ (x+y-1)^3 = x^3+y^3, \] and prove that they meet the curve only at infinity. Prove also that there is no point on the curve for values of \(x\) between \(a\) and \(1\), where \(a\) is the real root of the equation \[ 3a^3+3a^2-3a+1=0. \] Give a sketch showing the general form of the curve.