If f(x) = 3a|4x – 4| – ax, where a is some constant, find f ′(1)
with the answer options being:
0
not enough information
e
1

Question

If f(x) = 3a|4x – 4| – ax, where a is some constant, find f ′(1)
with the answer options being:
0
not enough information
e
1

misty1212 · Answer

HI!!

misty1212 · Answer

i would say "none of the above" since it is unlikely that this function has a derivative at $x=1$

misty1212 · Answer

suppose for example $a=1$ and you had $$f(x)=3|4x-4|-x$$ what would the derivative of that be?

anonymous · Answer

$$f \prime \left( x \right)~does~\not~exist~if~4x-4=0~or~x=1$$

michele_laino · Answer

if we write, explicitly, the absolute value, we can rewrite the function as below:
$$\Large f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  {12a\left( {x - 1} \right) - ax,\quad x > 1} \ 
  { - a,\quad x = 1} \ 
  {12a\left( {1 - x} \right) - ax,\quad x < 1} 
\end{array}} \right.$$
now, in order to see if it exists the function $f'(1)$, please you have to compute these two limits:
$$\Large \begin{gathered}
  \mathop {\lim }\limits_{x \to 1 - } \frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}} = ...? \hfill \
   \hfill \
  \mathop {\lim }\limits_{x \to 1 + } \frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}} = ...? \hfill \ 
\end{gathered} $$
of course, in order to do the substitution, you have to use my defintion above of the function