Show that the set of differentiable real-valued functions $f$ on the interval $(-4, 4)$ such that $f'(-1) = 3f(2)$ is a subspace of $\mathbf R^{(-4,4)}$.

Verifying the conditions in 1.34

• $0'(-1) = 3\cdot 0(2) = 0$
• $(f+g)'(-1) = f'(-1) + g'(-1) = 3f(2) + 3g(2) = 3(f + g)(2)$
• $(\lambda f)'(-1) = \lambda f'(-1) = \lambda 3f(2) = 3(\lambda f)(2)$