question_answer

If \[y=|x-{{x}^{2}}|,\] then find \[\frac{dy}{dx}\] at x = 1.

Answer:

We have, \[y=\,\,|x-{{x}^{2}}|\,\,=\left\{ \begin{align}   & x-{{x}^{2}},\,\,\text{if}\,\,0\le x\le 1 \\  & {{x}^{2}}-x,\,\,\text{if}\,\,0>x,\,\,x>1 \\ \end{align} \right.\] At x = 1, \[LHD={{\left[ \frac{d}{dx}(x-{{x}^{2}}) \right]}_{at\,\,x\,=\,1}}=1-2=-\,1\] \[RHD={{\left[ \frac{d}{dx}({{x}^{2}}-x) \right]}_{at\,\,x\,=\,1}}=2-1=1\] \[\therefore \]      \[LHD\ne RHD\] Hence, \[\frac{dy}{dx}\,\,at\,\,x=1\] does not exist.