Define a function $f(x)$ such that:
\begin{cases} \exp(-1/x^2), & \text{if } x>0, \\ 0, & \text{if } x\leqslant 0. \\ \end{cases}
I want to inductively prove that $f^{(n)}(0)=0$ for all $n \geqslant 0$. Any suggestions, please?
Define a function $f(x)$ such that:
\begin{cases} \exp(-1/x^2), & \text{if } x>0, \\ 0, & \text{if } x\leqslant 0. \\ \end{cases}
I want to inductively prove that $f^{(n)}(0)=0$ for all $n \geqslant 0$. Any suggestions, please?
We will show by induction that for each integer $n$, there exists a polynomial $P_n$ such that if $x\neq 0$ then $f^{(n)}(x)=P_n\left(\frac 1x\right)\exp\left(-\frac 1{x^2}\right).$ Put $P_0(X)=1$ and if $P_n$ works, then \begin{align} f^{(n+1)}(x)&=-\frac 1{x^2}P'_n\left(\frac 1x\right) \exp\left(-\frac 1{x^2}\right)+P_n\left(\frac 1x\right)\frac 2{x^3}\exp\left(-\frac 1{x^2}\right)\\ &=\left(\frac 2{x^3}P_n\left(\frac 1x\right)-\frac 1{x^2}P'_n\left(\frac 1x\right)\right)\exp\left(-\frac 1{x^2}\right). \end{align} We define $P_{n+1}(X)=2X^3P_n(X)-X^2P'_n(X),$ which is well-defined and completes the induction.