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

### Problem 4
- \( f(x) \) is twice differentiable on \( \mathbb{R} \), and \( M_k = \sup |f^{(k)}(x)| < +\infty \) for \( k = 0, 1, 2 \).

### Problem 5
- Given \( \lambda > 0 \) and \( 0 < \alpha < \beta \), there exists \( |\theta| < 1 \) such that:
  \[
  \alpha \int_{\alpha}^{\beta} \frac{e^{-\lambda x}}{x} \cos x \, dx = 2\theta.
  \]

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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.

Explore advanced mathematical problems involving twice-differentiable functions with bounded derivatives and the evaluation of definite integrals combining exponential and trigonometric terms. Suitable for advanced undergraduate or graduate students.

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