> [!NOTE] Corollary (Functions with zero derivative) > If $f:I \to \mathbb{R}$ is differentiable on the open interval $I$ and $f'(x)=0$ for all $x\in I$ then $f$ is constant on the interval. >*Proof*. [[Mean value theorem]] # Examples We can use this lemma to prove uniqueness of solutions for [[Ordinary Differential Equation|DEs]]. > [!Example] Uniqueness of solution to a Diff. Eq. I > The only functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f'(x)=f(x)$ are the functions $f(x)=Ae^{x}$ for some constant $A$. > > **Proof** > We shall assume $e^{x}$ is differentiable and is its own derivative. > Suppose $f$ is such a solution and let $g(x) = e^{-x}f(x)$. Then by [[Chain rule for derivative|chain rule]], $g'(x)=-e^{x}f(x) + e^{-x}f'(x) = e^{-x}(-f(x)+f(x))=0$Since [[Real Function with Zero Derivative is Constant]] $g$ is a constant function with value say $A$. > Then $f(x)=e^{x}g(x)=Ae^{x}$. > > [!Example] Uniqueness of solution to a Diff. Eq. II > The only functions $f:(0, \infty)\to \mathbb{R}$ satisfying $f'(x)+\frac{1}{x}f(x)=2$are the functions $f(x)=\frac{A}{x}+x$ for some $A\in\mathbb{R}$. > >**Proof** >Exercise >' Other examples: - [[Real Exponential Function of Sum]]; - [[Trigonometric Functions|Compound angle formulae]];