Show that there is a unique pair of real numbers \(a\), \(b\) with the property that \[\int_{-1}^{+1} P(x) dx = P(a) + P(b)\] for all polynomials \(P(x)\) of degree at most three.
Let \(a_1, \ldots, a_n\) be \(n\) real numbers such that \(0 > a_i \geq -1\) for each \(i\). Prove that $$(1+a_1)\ldots(1+a_n) > 1+a_1+\ldots+a_n$$ if \(s > 1\).
Let \(\Gamma\) be an ellipse in the \((x, y)\) plane, whose axes are not necessarily parallel to the coordinate axes and whose centre is not necessarily at the origin. Let \(V\) be the set of points inside or on \(\Gamma\). Show that as \(z = x+iy\) varies over \(V\), with \(z_0\) a fixed complex number, \(|z-z_0|\) reaches its maximum value when \(z\) is on \(\Gamma\). If \(\Gamma\) is a circle of radius \(r\) with centre at the origin, find the point \(z\) of \(\Gamma\) such that \(|z-z_0|\) has its maximum value.
Find \(\displaystyle \sum_{n=0}^N n\cos n\theta\). Prove that this series does not converge as \(N\) tends to infinity, for any given real value of \(\theta\).
Find a real value of \(x\) making $$f(x) = -3|x|^4 + 8|x|^3 + 6|x|^2 - 24|x| - 201$$ as large as possible. A proof that any other real value of \(x\) gives a smaller value of \(f(x)\) should be included.
Let \(d\), \(e\), \(f\) and \(g\) be fixed integers. Let $$ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g = 0$$ have at least three roots, not necessarily distinct, each of which is a non-zero integer. Prove that \(a\), \(b\) and \(c\) are rational numbers.
(i) Evaluate $$\int_0^{\frac12\pi} x\left(\tfrac12\pi - x\right)\sin^2 x \, dx.$$ (ii) Find the general solution of the differential equation $$(x^2 \log x)y' + xy = (x^2 \log x - 1)\cos x$$ in the range \(x > 1\).
The function \(f(x)\) is said to be maximal in the closed interval \([a, b]\) at \(c\) if (i) \(a \leq c \leq b\) and (iii) \(f(x) \leq f(c)\) whenever \(a \leq x \leq b\). If \(f(x)\) is maximal in \((a, b)\) at \(c\), where \(a < c < b\), and \(f'(c)\) exists, show that \(f'(c) = 0\). You may assume the theorem that a function continuous in a closed interval is maximal in that interval at at least one point. Suppose that \(f(x)\) is continuous in \([a, b]\) and that \(f(x) < g(x)\) for all \(x\) such that \(a < x < b\). Show that, if \(a < y < b\) then \(f(x) \leq f(x)\) and deduce that \(f(x)\) is maximal in \([a, b]\) at \(b\) and nowhere else.
If \(g(x)\) has a continuous \(n\)th derivative, and satisfies $$g(0) = g'(0) = g''(0) = \ldots = g^{(n-1)}(0) = 0,$$ prove that $$g(x) = \frac{1}{(n-1)!} \int_0^x g^{(n)}(t)(x-t)^{n-1} dt.$$ Deduce that, if \(f(x)\) has a continuous \(n\)th derivative, $$f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + \ldots + \frac{x^{n-1}}{(n-1)!}f^{(n-1)}(0) + R_n(x),$$ where $$R_n(x) = \frac{1}{(n-1)!} \int_0^x f^{(n)}(t)(x-t)^{n-1} dt.$$ If now \(f(x) = (1+x)^{1/2}\), show that \(|R_n(x)| < 1/(n-1)\) for all \(n > 2\) and all \(x\) such that \(-1 < x \leq 1\). What conclusion do you draw from this result?
If \(\theta(t)\) and \(\phi(t)\) are differentiable functions of an independent variable \(t\), and \(F(t) = f(\theta(t), \phi(t))\), where \(f\) has continuous first-order partial derivatives, prove that $$\frac{dF}{dt} = \frac{\partial f}{\partial \theta} \frac{d\theta}{dt} + \frac{\partial f}{\partial \phi} \frac{d\phi}{dt}.$$ The variables \(x\), \(y\), \(z\), \(t\) are such that any two can be regarded as independent, and the other two can then be expressed as functions of them. The partial differential coefficients of \(x\) regarded as a function of \(y\) and \(z\), are continuous and are denoted by \((\partial x/\partial y)_z\) and \((\partial x/\partial z)_y\) and others likewise. Prove that $$\left(\frac{\partial x}{\partial y}\right)_z = \left(\frac{\partial x}{\partial t}\right)_z \left(\frac{\partial t}{\partial y}\right)_z = \left(\frac{\partial x}{\partial y}\right)_t + \left(\frac{\partial x}{\partial t}\right)_y \left(\frac{\partial t}{\partial y}\right)_z.$$ A chord of length \(l\) subtends an angle \(\theta(0 < \theta < \pi)\) at the centre of a circle of radius \(r\); the area of the smaller of the segments into which the circle is dissected by the chord is \(A\). Express \((\partial A/\partial r)_l\) and \((\partial A/\partial l)_r\) as functions of \(r\) and \(\theta\).