Search for question
Question

\begin{aligned} &\text { Problem 3. Let } \mathcal{D}=\{f: \mathbf{R} \rightarrow \mathbf{R} \mid f \text { is differentiable (at all } x \in \mathbf{R} \text { )\}. For }\\ &f \in

\mathcal{D}, f^{\prime}: \mathbf{R} \rightarrow \mathbf{R} \text { is the derivative of } f . \text { Define a binary relation } E \text { on } \mathcal{D} \text { by } \end{aligned} f E g \Longleftrightarrow f^{\prime}=g^{\prime} (a) Prove that E is an equivalence relation. (b) Which functions are in the same equivalence class as the function f : R →R defined by f (x) = x²?

Fig: 1

Fig: 2

Fig: 3

Fig: 4