3 Show that the set of differentiable real-valued functions 𝑓 on the interval (−4, 4) such that 𝑓'(−1) = 3 𝑓 (2) is a subspace of $𝐑^{-4,4}$. SOLUTION Let $𝑈 = \{𝑓 ∈ 𝐑^{(-4,4)}$ ∶ 𝑓 is differentiable and 𝑓' (−1) = 3𝑓(2)}. Clearly the 0 function is in 𝑈. The sum of any two differentiable functions is differentiable, as is every constant multiple of any differential function. Suppose 𝑓,𝑔 ∈ 𝑈 and 𝑐 ∈ 𝐑. Then $ \begin{align} (f+𝑔)'(−1)&= 𝑓'(−1)+𝑔'(−1) \\ &= 3 𝑓 (2) + 3 𝑔(2)\\ &= 3( 𝑓 + 𝑔)(2)\\ \end{align} $ and $\begin{align} (c 𝑓 )'(−1) &= 𝑐 𝑓 '(−1) \\ &= 3𝑐𝑓(2\\ &= 3(𝑐 𝑓 )(2). \\ \end{align} $ Thus 𝑓 + 𝑔 ∈ 𝑈 and 𝑐 𝑓 ∈ 𝑈. Thus 𝑈 satisfies the three conditions in 1.34 and hence 𝑈 is a subspace of $R^{(−4,4)}$. ---