Slide
This function involves absolute values. To find the slope on the
slide use different equations to define the function in different
parts of its domain.
Age
16 to 18
Challenge level
Problem
Plot the graph of the function $y=f(x)$ where $f(x) = x(x+|x|)$. Find the first and second derivatives of the function. Show that the first derivative exists at $x=0$ but that the second derivative does not exist at $x = 0$.
NOTES AND BACKGROUND
This is a very simple question but it requires an understanding of how to handle functions that are defined differently on different parts of their domains.
Getting Started
Split up the domain of the function.
Student Solutions
This solution was written by Andrei from Tudor Vianu National College, Bucharest, Romania.
I calculate the values of $f(x)= x(x+|x|)$ for $x< 0$ and $x\geq 0$:
$$\eqalign{ f(x) &= x(x-x)=0\ for \ x < 0 \cr &=x(x+x)= 2x^2 \ for \ x\geq 0.}$$
Its graph is represented below:
Image
Now, I find the first derivative of $f(x)$:
$$\eqalign{ f'(x) &= 0\ {\rm for}\ x < 0 \cr &= 4x
\ {\rm for}\ x\geq 0}$$
I observe that for $x = 0$, $f'(x)$ is $0$ both from the first
and from the second form of $f(x)$, that is on both sides of the
origin. So the first derivative $f'(0)=0$ exists at $x=0$. Hence
the graph of $f(x)$ has the tangent at $y=0$ at the origin.
Now, I calculate the second derivative:
$$\eqalign{ f''(x) &= 0\ for\ x < 0 \cr &= 4\ for \
x> 0.}$$
Hence the second derivative does not exist at the origin
because on the left the limiting value of $f''(x)$ as $x\to 0$ is
$0$ whereas on the right the limiting value of $f''(x)$ as $x\to 0$
is $4$. So there isn't a unique tangent to the graph of $f'(x)$ at
$x = 0$.
Teachers' Resources
Why do this problem?
It requires an understanding of how to handle functions that are defined differently on different parts of their domains and how to interpret the situation when the derivative takes different values close to point but on opposite sides of the point.
Possible approach
A short problem that can be used as a lesson starter.
Key questions
What can you say about the function when $x< 0$?
What can you say about the function when $x> 0$?