# find critical numbers of function with absolute value

• May 18th 2012, 12:47 AM
rabert1
find critical numbers of function with absolute value

$g(t) = |3t-4|$
• May 18th 2012, 12:54 AM
Goku
Re: find critical numbers of function with absolute value
Draw the graph, then find the "V" point or cusp and that will be your critical point.
• May 18th 2012, 01:06 AM
rabert1
Re: find critical numbers of function with absolute value
Can you kindly show how to do this with calculus\algebra
• May 18th 2012, 01:19 AM
Goku
Re: find critical numbers of function with absolute value
• May 18th 2012, 01:22 AM
rabert1
Re: find critical numbers of function with absolute value
nice find. thanks
• May 18th 2012, 09:04 AM
HallsofIvy
Re: find critical numbers of function with absolute value
You start with definitions! A "critical point" of a function of one variable is a point where either the derivative is 0 or the derivative does not exist.
The |x| is x if $x\ge 0$ and -x if $x< 0$.

In particular |3t- 4| is 3t- 4 if $3t- 4\ge 0$ or $t\ge 4/3$, -(3t- 4)= 4- 3t for [tex]t< 4/3[/tex[. So for [tex]t> 4/3[tex], the function is 3t- 4 and has derivative 3. For $x< 3/4$, the function is 4- 3t which has derivative -3. Those are never 0 so the only possible critical point would be at x= 4/3. We need to look at that point separately.

To determine the derivative at x= 4/3 we can do either of two things:

1) Use the basic definition. The derivative of f(x) at x= 4/3 is $\lim_{h\to 0}\frac{f((4/3)+h)- f(4/3)}{h}$. f(4/3)= 0, of course, but because of the change of formula on both sides of x= 4/3, we need to look at the "one sided" derivatives. If h> 0 then (4/3)+ h> 4/3, f((4/3)+ h)= 3((4/3)+ h)- 4= 3h so the difference quotient is $\frac{3h}{h}= 3$ and the limit is 3. If h< 0 then (4/3)+ h< 4/3, f((4/3)+h)= 4- 3((4/3)+ h)= -3h so the difference quotient is $\frac{-3h}{h}= -3$ and the limit is -3. Since those two one sided limits are not the same, the limit itself does not exist and the function is not differentiable at x= 4/3.

2) Use a more "sophisticated" theorem. While the derivative of a differentiable function is not itself necessarily continuous, one can use the mean value theorem to show that the "intermediate value theorem" does, in fact, apply. From that, it follows that if the derivative at a specific point exists, the two limits $\lim_{x\to a^-}f'(x)$ and $\lim_{x\to a^+}f'(x)$ must both be equal to the derivative at that point. Here, of course, those two limits are -3 and 3 and are not the same so the function is not differentiable at x= 4/3.