Integral inequality
An inequality involving integrals of squares of functions.
Problem
(i)Suppose that $a$, $b$ and $t$ are positive. Which of
the following two expressions is the larger (ii)By considering
the inequality prove
that, for all functions $f(x)$ and $g(x)$,
Getting Started
The first part of this question involves an easy evaluation of the
integrals and then some simple algebra to compare the
results.
In the second part you will get a quadratic inequality in $\lambda$ of the form Completing the
square leads to a condition for this inequality to hold which gives
the required result.
In the second part you will get a quadratic inequality in $\lambda$ of the form
Student Solutions
Shaun from Nottingham High School sent this solution.
(i) By calculation, we have
(ii) Given the inequality
Teachers' Resources
Here the first part is a special case of the result in the second
part. However it is easier to prove the general case
directly.