Evaluate the integral $$I_n = \int_0^{\pi/2} \sin^n x \, dx$$ by using a reduction formula. By comparing the integrals \(I_{2n}\), \(I_{2n+1}\) and \(I_{2n+2}\), or otherwise, show that $$\frac{4}{3} \cdot \frac{16}{15} \cdots \frac{4n^2}{4n^2 - 1} \to \frac{\pi}{2} \quad \text{as} \quad n \to \infty.$$
Sketch the cubic curve $$(xy - 12)(x + y - 9) = a$$
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 $$\lambda x + \mu y \geq x^{\lambda} y^{\mu}.$$