COURSE: Mathematical Analysis/Real Analysis (CC1A)TOPIC: Continuity + Connectedness **Prove $ f(x) $ has a zero (i.e., a point where $ f(p) = 0 $) on each interval. You can assert that the functions are continuous on the relevant intervals (i.e., a rigorous proof that $ f(x) $ is continuous on the given interval is not needed).**\[ f(x) = 2e^{-x} (\cos(2x)); \text{ on } \left[0, \frac{\pi}{2}\right] \]---### Explanation:The function $ f(x) = 2e^{-x} (\cos(2x)) $ is defined on the interval $[0, \frac{\pi}{2}]$. The goal is to demonstrate that a zero exists within this interval, meaning there exists a point $ p $ within $[0, \frac{\pi}{2}]$ where $ f(p) = 0 $. Continuity within the interval can be assumed without a detailed proof due to the nature of exponential and trigonometric functions composing $ f(x) $.

We are given a function f(x) = 2e-x (cos(2x)). We need to prove that f(x) has a zero on each…

Prove f(x) has a zero (i.e., a point where f(p)=0) on each interval. You can assert that the functions are continuous on the relevant intervals (i.e., a rigor- ous proof that f(x) is continuous on the given interval is not needed). f (x) = 2e-"(cos(2x)); on [0, 7] %3D

Prove f(x) has a zero (i.e., a point where f(p)=0) on each interval. You can assert that the functions are continuous on the relevant intervals (i.e., a rigor- ous proof that f(x) is continuous on the given interval is not needed). f (x) = 2e-"(cos(2x)); on [0, 7] %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

COURSE: Mathematical Analysis/Real Analysis (CC1A)

TOPIC: Continuity + Connectedness

**Prove $ f(x) $ has a zero (i.e., a point where $ f(p) = 0 $) on each interval. You can assert that the functions are continuous on the relevant intervals (i.e., a rigorous proof that $ f(x) $ is continuous on the given interval is not needed).**

\[ f(x) = 2e^{-x} (\cos(2x)); \text{ on } \left[0, \frac{\pi}{2}\right] \]

---

### Explanation:

The function $ f(x) = 2e^{-x} (\cos(2x)) $ is defined on the interval $[0, \frac{\pi}{2}]$. The goal is to demonstrate that a zero exists within this interval, meaning there exists a point $ p $ within $[0, \frac{\pi}{2}]$ where $ f(p) = 0 $. Continuity within the interval can be assumed without a detailed proof due to the nature of exponential and trigonometric functions composing $ f(x) $.

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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