Problems

Sketch the cubic curve $$(xy - 12)(x + y - 9) = a$$

1. [(i)] for a small positive value of the constant \(a\), and
2. [(ii)] for a small negative value of the constant \(a\).

For what value of \(a\) does the curve have an isolated point?

The function \(f(x)\) is such that \(f'(t) \geq f'(u)\) whenever \(t \leq u\). By applying the Mean Value Theorem to the function \(f\) over suitable intervals, or otherwise, show that $$f(\lambda x + \mu y) \geq \lambda f(x) + \mu f(y)$$ whenever \(\lambda \geq 0\), \(\mu \geq 0\), \(\lambda + \mu = 1\). By taking a suitable function \(f\) (or otherwise) show that if \(x\), \(y\) are positive and \(\lambda\), \(\mu\) are as above, we have $$\lambda x + \mu y \geq x^{\lambda} y^{\mu}.$$