Find the maximum and minimum values of \[ (x+3)^2(x-2)^3, \] and draw a rough graph of the function. Hence, or otherwise, prove that the equation \[ x^5 - 15x^3 + 10x + 60 = 0 \] has two real negative roots and two imaginary roots.
Integrate
Find the area of the loop of the curve \[ a^3y^2 = x^4(2x+a). \]
\(ABC\) is a triangle inscribed in a circle. \(AP\) is a chord of the circle which bisects \(BC\), and the tangent at \(P\) meets \(BC\) in \(N\). Prove that \[ NC:NB = AB^2:AC^2. \]
\(OP, OQ\) are tangents at \(P, Q\) to a parabola, and the line bisecting \(PQ\) at right angles meets the axis in \(G\). Prove that the focal chord parallel to \(PQ\) bisects \(OG\).
Shew that it is possible for two perpendicular normal chords of an ellipse to meet on the curve if \(e^2 \ge \frac{2}{3}\). Shew also that in this case each of the chords is trisected by another perpendicular normal chord.
\(OBP, OAQ\) are the asymptotes of a conic, \(A, B\) being fixed points and \(PQ\) a variable tangent. Prove that \(PA, QB\) will intersect on a conic which has parallel asymptotes and passes through \(A, B\).
Prove that the foci of the hyperbola \(xy - 2ax - 2by + 2a^2 = 0\) lie on one or other of the parabolas \(y^2 = 4ax, y^2 = 4a(2a-x)\).
Shew that, if \(a, b, c, x, y, z\) denote real numbers, and the sum of any two of the three \(a, b, c\) is greater than the third, then \[ a^2(x-y)(x-z) + b^2(y-x)(y-z) + c^2(z-x)(z-y) \] is positive.
Prove that, if \(N\) and \(n\) are nearly equal, then \[ \left(\frac{N}{n}\right)^{1/3} = \frac{N+n}{n + \frac{1}{3}\frac{N+n}{4n-N}} \text{ approximately,} \] the error being approximately \(\frac{7}{81}\left(\frac{N-n}{n}\right)^3\).