If f(x) = 3a|4x – 4| – ax, where a is some constant, find f ′(1). Please help me guys, I give medals @preethat @phi

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If f(x) = 3a|4x – 4| – ax, where a is some constant, find f ′(1). Please help me guys, I give medals @preethat @phi

Mathematics
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@phi could you help me
I remember this question and there was something fishy about it.. if a=0 then f(x)=0 then f'(x)=0 and so f'(1)=0 but what if a is not 0 then we have to consider 4x-4>0 and 4x-4<0 if 4x-4>0 then x-1>0 and then so f(x)=3a(4)(x-1)-ax if x>1 find f' for this side and if 4x-4<0 then x-1<0 and then so f(x)=3a(-4)(x-1)-ax if x<1 find f' for this one see if f'(1) matches for both sides
  • phi
the derivative of | f(x) | is f(x) f'(x)/|f(x)|

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I say it was fishy because of the choices that came with this problem
something about A) 1 B) 0 C) -1 D) can't remember
I don't know if those were the choices actually
Yeah, but could you explain me what they are actually asking, I mean I not want just the answer i was to know how to get answer because I really want to pass this exam
they are asking for f'(1)
I broken into a piecewise function
so that means that first I have to find the derivate function and then plug 1?
@phi gave you something where you don't need piecewise function
yes
  • phi
f(x) = 3a|4x – 4| – ax, or f(x)= 12a | x -1| -ax the derivative is 12 a (x-1)/| x-1| - a
hopefully the a is not zero right it should really specify that the constant is not zero
  • phi
notice at x=1 we have an undefined quantity and the best we can do is write a limit, which is different for 1- and 1+
well that already says f' doesn't exist at x=1 we don't have to do the limit thing
those are the only options, but I am not sure whether to pick 1 or not enough 0 not enough information e 1
I would say not enough info because we don't know if a is 0 or not
like phi just show the f'(1) doesn't exist when a isn't 0 but when a=0 ,f(x)=0 and so f'(x)=0 and so f'(1)=0
you get two difference answers for two difference cases of a
true, I will select it and I'll see if the answer is correct, @phi said something similar, but i don't think they gave me enough information to solve the problem
is that a modulus or a bracket?
if you want me to explain the piecewise function more I can by the way @Joseluess
if you want
yes please, I am a little confused
modulus @14mdaz
\[|x|=\left[\begin{matrix}x & \text{ if } x>0 \\ -x & \text{ if } x<0 \\ 0 & \text{ if } x=0\end{matrix}\right]\] sorry don't know how to type a pretty piecewise function had to use the matrix code have you ever seen this and do you understand this?
yes I know what you're talking about
|-2|=-(-2) I used row 2 since x=-2<0 |2|=2 I used row 1 since x=2>0 |0|=0 used row 3 since x=0 anyways just in case you didn't know that now I could replace all those x's with any function and solve any resulting inequalities that occur for example I see we have |4x-4| so I'm going to replace all the x's with (4x-4)'s
\[|4x-4|=\left[\begin{matrix}4x-4 & \text{ if } 4x-4>0 \\ -(4x-4) & \text{ if } 4x-4<0 \\ 0 & \text{ if } 4x-4=0\end{matrix}\right]\]
now straightening this up a bit \[|4x-4|=\left[\begin{matrix}4x-4 & \text{ if } x>1 \\ -(4x-4) & \text{ if } x<1 \\ 0 & \text{ if } x=1\end{matrix}\right] \]
but when x>1 what is the derivative of |4x-4| aka 4x-4 since |4x-4|=4x-4 for when x>1
\[(|4x-4|)'=\left[\begin{matrix}(4x-4)' & \text{ if } x>1 \\ -(4x-4)' & \text{ if } x<1 \\ ? & \text{ if } x=1\end{matrix}\right] \\ =...\] I left that one thing as a ? because we have to see if left derivative=right derivative (as x approaches 1 from both sides)
\[(|4x-4|)'=\left[\begin{matrix}(4x-4)' & \text{ if } x>1 \\ -(4x-4)' & \text{ if } x<1 \\ ? & \text{ if } x=1\end{matrix}\right] \\ (|4x-4|)'=\left[\begin{matrix}4 & \text{ if } x>1 \\ -4 & \text{ if } x<1 \\ \text{ does not exist (since*)} & \text{ if } x=1\end{matrix}\right] \\ \\ \text{ since } f'(1^+) \ \neq f'(1^-) \\ \text{ that is } 4 \neq -4 \]
now this already tells you that your function f(x)=3a|4x-4|-ax if a isn't 0 that you have f'(1) doesn't exist because (|4x-4|)' evaluated at x=1 doesn't exist but now if a=0 your function is f(x)=3(0)|4x-4|-0x=0-0=0 your function is f(x)=0 and so f'(x)=0 since derivative of a constant is 0 now f'(x)=0 does say it doesn't matter what x we have the f' value will always be 0 so f'(1)=0 if a=0
so summary of it all f'(1) can be 0 or doesn't exist
the information that doesn't allow us to say which is all dependent on what a is
Okay, so that's why the answer it not enough right? okay I understand it better thanks
np

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