Math Problem Statement

Problem 4: f(x) is twice differentiable on R, and Mk = sup |f^(k)(x)| < +∞ for k = 0, 1, 2. Problem 5: Given λ > 0 and 0 < α < β, there exists |θ| < 1 such that: α ∫[α, β] (e^(-λx)/x) cos(x) dx = 2θ.

Solution

The image contains two mathematical problems. Let me transcribe and analyze each:

Problem 4

  • f(x)f(x) is twice differentiable on R\mathbb{R}, and Mk=supf(k)(x)<+M_k = \sup |f^{(k)}(x)| < +\infty for k=0,1,2k = 0, 1, 2.

Problem 5

  • Given λ>0\lambda > 0 and 0<α<β0 < \alpha < \beta, there exists θ<1|\theta| < 1 such that: ααβeλxxcosxdx=2θ.\alpha \int_{\alpha}^{\beta} \frac{e^{-\lambda x}}{x} \cos x \, dx = 2\theta.

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Math Problem Analysis

Mathematical Concepts

Real analysis
Differentiability
Definite integrals
Exponential functions
Trigonometric functions

Formulas

Mk = sup |f^(k)(x)|, for k = 0, 1, 2
α ∫[α, β] (e^(-λx)/x) cos(x) dx = 2θ

Theorems

Properties of differentiable functions
Supremum definition
Integration by parts for definite integrals

Suitable Grade Level

Advanced undergraduate or graduate level