MathDB

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Part of 2024 VJIMC

Problems(2)

Inequality of function with integral and derivative

Source: VJIMC 2024, Category I, Problem 1

4/14/2024
Let f:R→Rf:\mathbb{R} \to \mathbb{R} be a continuously differentiable function. Prove that ∣f(1)−∫01f(x)dx∣≤12max⁡x∈[0,1]∣f′(x)∣.\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.
inequalitiesfunctioncalculusintegrationderivative
Cauchy-like inequality implies that function has a root

Source: VJIMC 2024, Category II, Problem 1

4/14/2024
Suppose that f:[−1,1]→Rf:[-1,1] \to \mathbb{R} is continuous and satisfies (∫−11exf(x)dx)2≥(∫−11f(x)dx)(∫−11e2xf(x)dx).\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right). Prove that there exists a point c∈(−1,1)c \in (-1,1) such that f(c)=0f(c)=0.
inequalitiesfunctionexponentialcalculus