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Age 16 to 18
Challenge Level
(i)Suppose that $a$, $b$ and $t$ are positive. Which of the following two expressions is the larger $$P=\left(\int_0^t x^{a+b}dx\right)^2, \qquad Q=\left(\int_0^t x^{2a}dx \right) \left(\int_0^t x^{2b}dx\right)\ ?$$ (ii)By considering the inequality $$\int_0^t [f(x)+\lambda g(x)]^2 dx \geq 0,$$ prove that, for all functions $f(x)$ and $g(x)$, $$\left(\int_0^t f(x)g(x)dx\right)^2 \leq \left(\int_0^t f(x)^2 dx\right) \left(\int_0^t g(x)^2 dx\right).$$