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Problem 60 Hard Difficulty

Determine whether $ f'(0) $ exists.

$ f(x) = \left\{
\begin{array}{ll}
x^2 \sin \frac{1}{x} & \mbox{if $ x \neq 0 $}\\
0 & \mbox{if $ x = 0 $}
\end{array} \right.$

Answer

$f^{\prime}(0)=0$

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Video Transcript

we have a question in which we need to get a mind whether actors directors are not. So we have effects which is equal to access squired sign one by X. When X is not equal to zero and zero of annex equal to zero. Okay, so we need to find um we need to differentiate first and we need to find left and innovative. So if differentiation if derivative exists so left and derivatives should be able to write and innovative. So left and derivative will be Limit. X approaches to zero from left side, affects minus, affects minus f zero by x minus zero. 40 by X zero. No, if you convert it into which it will become Limit accidents to zero F zero minus set plus f zero by zero minus at minus zero, 0- at minus F zero x 0- set minus zero as we are approaching X from the left side. Okay, so this is Limit at approaches to zero. F off minus set -F of zero, divided by minus that. So let us plug in the value in the function at approaches to zero F minus set is minus that whole square Sign -1 by H zero because 0 is zero. My mindset. So this will become limited accidents to zero at a square Sign -1 by H by -H. So I wanted to and actually get canceled out. We'll be getting and one more thing sign of any negative number sign -1 battled mind sign it. So this limit as approaches to 0- at sign But one by H Divide by- at -1. Yeah. Okay so finally we'll be getting at sign one batch now if that approaches to zero. Uh So this will become zero into Sign one by H. So we know that range of signage. A range of sine function is always -1-1. So any value of one batch will give us between any number between -1-1. So this is zero. So left unlimited zero similarly right different derivatives right And elevating will be limit at approaches to zero from right side, F X -F0 By X zero. So this will be limit. Sorry here actually approaching 20 limit Such approaches to zero, plus H minus f zero but express edge. Uh It should not be a plus at it should be zero plus at Yeah. Okay. Zero blacks. Okay in the denominator it should be zero plus eight minus zero. Triple A. Search zero. So this will be a limit. That approach is 20 F h minus F zero by H. Now plugging in the function Limited approaches to zero and 2 square sign one by it and F 00. So let it be much. Now again this will tend to zero and this will tend Any value between my 1-10. So we can see that left and derivative equal to write and debating hands. The limit exists. Thank you him limited. Very very big. Sorry, thank you.