Question

If y = |x − x²|, then find $\frac{d^{2}y}{d{x}^2}$.

Answer 1

Here,
$$\begin{gathered} y=\left|x-x^2\right| \\ =\left\{\begin{array}{l}x-x^2\mathrm{if}0<x<1\\ -x+x^2\mathrm{if}x>1,x<0\end{array}\right. \\ \Rightarrow \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\left\{\begin{array}{l}1-2x\mathrm{if}0<x<1\\ -1+2x\mathrm{if}x>1,x<0\end{array}\right. \\ \Rightarrow \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}=\left\{\begin{array}{l}-2\mathrm{if}0<x<1\\ 2\mathrm{if}x>1,x<0\end{array}\right. \end{gathered}$$