Mark all true statements (there might be more than one statement that is true). O Let f: AC R→ R and assume that, for all x, ye A, f(x) = f(y). Then for all c E Int(A), f'(c) = 0. x²sin :) if x # 0 Let f: R→ R be given by f(x) = { 0 Let f: (a, b) → R be differentiable on (a,b) and assume that f' is continuous at c € (a, b). If f'(c) >0 then there is > 0, such that for all x, y = D(c,d) n (a,b), if x
Mark all true statements (there might be more than one statement that is true). O Let f: AC R→ R and assume that, for all x, ye A, f(x) = f(y). Then for all c E Int(A), f'(c) = 0. x²sin :) if x # 0 Let f: R→ R be given by f(x) = { 0 Let f: (a, b) → R be differentiable on (a,b) and assume that f' is continuous at c € (a, b). If f'(c) >0 then there is > 0, such that for all x, y = D(c,d) n (a,b), if x
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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Transcribed Image Text:**Question 1**
Mark all true statements (there might be more than one statement that is true).
- □ Let \( f : A \subseteq \mathbb{R} \rightarrow \mathbb{R} \) and assume that, for all \( x, y \in A \), \( f(x) = f(y) \). Then for all \( c \in \text{Int}(A) \), \( f'(c) = 0 \).
- □ Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be given by \( f(x) = \begin{cases} x^2 \sin \left( \frac{1}{x} \right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \). Then \( f' \) is continuous.
- □ Let \( f : (a, b) \rightarrow \mathbb{R} \) be differentiable on \( (a, b) \) and assume that \( f' \) is continuous at \( c \in (a, b) \). If \( f'(c) > 0 \) then there is \( \delta > 0 \), such that for all \( x, y \in D(c, \delta) \cap (a, b) \), if \( x < y \) then \( f(x) < f(y) \).
- □ Let \( f, g: A \subseteq \mathbb{R} \rightarrow \mathbb{R} \) be differentiable for all \( x \in \text{Int}(A) \) and \( f'(x) = g'(x) \), for all \( x \in \text{Int}(A) \). Then there is \( C \in \mathbb{R} \), such that \( f(x) = g(x) + C \).
- □ Let \( f: A \subseteq \mathbb{R} \rightarrow \mathbb{R} \) be differentiable for all \( x \in \text{Int}(A) \) and \( f(x) = 0 \). Then \( f \) is constant.
**Question 2**
Mark all true statements (there might be more than one statement that is true).
- □ Let
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