Integral inequality - École Polytechnique International Entrance Exam

Hint: You haven't really made a mistake as such. As you've already realised, you just haven't use all the pieces of information you've been asked to gather in parts $1.1$ and $1.2$. Besides not having used $1.2$, you also haven't used the second inequality of $(1)$. What was your answer to $(1.2)$? What is $\ \displaystyle\int_{\frac{a+b}{2}}^b(b-t)dt\ $? What is the sum of those two quantities?

Your first step in 1.3 is good. But you are not using 1.2. So I suggest this:

$$\int_a^b |f(t)|\,dt = \int_a^{(b+a)/2}|f(t)|\,dt + \int_{(b+a)/2}^b|f(t)|\,dt.$$

$$ \eqalign{ & \forall t \in \left[ {a,b} \right]\;:\;\left| {f(t)} \right| \le K(t - a)\; \wedge \;\left| {f(t)} \right| \le K(b - t)\; \Rightarrow \cr & \Rightarrow \;2\left| {f(t)} \right| \le K\left( {(t - a) + (b - t)} \right) \Rightarrow \quad \cdots \cr} $$