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--- pow:problem1f23 [2023/09/11 04:41] – created mazur
+++ pow:problem1f23 [2023/09/18 18:42] (current) – mazur
@@ Line 1: / Line 1: @@
+<box 85% round orange| Problem 1 (due Monday, September 11)>
+A continuous function $f:\mathbb R\longrightarrow \mathbb R$ has the following property:
+\[f(x)\cdot f(f(x))=1\ \text{for every}\ x\in\mathbb R.\]
+Knowing that the largest value of $f$ is $e$, prove that
+\[3+e^{-2}<\int_0^{2e}f(t)dt<3+e^2.\]
+Show that these bounds are best possible. Here $e=2.7128...$ is the base of natural logarithms.
+</box>
+We received solutions from Sasha Aksenchuk, Prof. Vladislav Kargin, Mithun Padinhare Veettil, and Daniel J. Riley (Tufts U.). All solvers provided a correct argument for the inequalities
+ \[3+e^{-2}\leq\int_0^{2e}f(t)dt\leq 3+e^2.\]
+The justification that the above inequalities are strict was usually not provided in sufficient detail and some
+solutions did not provide sufficiently detailed justification that the bounds are best possible. For a detailed solution
+see the following link {{:pow:2023fproblem1.pdf|Solution}}.