- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- Prove the zero-derivative theorem for vector-valued functions. If $F'(t) = O$ for each $t$ in an open interval $I,$ then there is a vector $C$ such that $F(t) = C$ for all $t$ in $I.$
- In problem (18), we showed that a vector-valued function is differentiable if and only if for every $t$ in an open interval $I,$ and for every component $f_i$ of $F$ $$f_i'(t) = \lim_{h \to 0}\frac{1}{h}\left[f_i(t + h) - f_i(t)\right]$$Now, suppose that for every $t$ in the open interval $I,$ $f_i'(t) = 0.$ Then, the above limit can be rewritten as $$\lim_{h \to 0}\frac{1}{h}\,f_i(t + h) = f_i(t)$$But this implies that there is some real scalar $c_i$ such that for every $t$ in $I,$ $f_i(t) = c_i.$ But if this is true for each $f_i$ of $F,$ then the vector-valued function $F(t) = C$ on the open interval $I,$ where $C = (c_1, ..., c_n). \quad \blacksquare$