You may also like

problem icon


Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

problem icon

Sine Problem

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.

problem icon

Cocked Hat

Sketch the graphs for this implicitly defined family of functions.


Age 16 to 18 Challenge Level:

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:
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 $x = 0$.