=for timestamp
Sa Mär  5 15:12:29 CET 2005

=head3 44. Hausaufgabe

=head4 Selbstgestellte Aufgabe

M<<
{} f(x) = x^3 - 3x; \quad D_f = \mathds{R}; \\
{} f'(x) = 3x^2 - 3 = 3\left(x + 1\right)\left(x - 1\right);
>>

M<<
{} f'(x) E<gt> 0; \Rightarrow 3\left(x + 1\right)\left(x - 1\right) E<gt> 0;
{} \Rightarrow \begin{cases}
{}   f'(x) E<gt> 0; \Rightarrow f \text{ ist sms}; &
{}     \text{f"ur } x \in \mathds{R} \setminus \left] -1, 1 \right[; \\
{}   f'(x) E<lt> 0; \Rightarrow f \text{ ist smf}; &
{}     \text{f"ur } x \in \left] -1, 1 \right[;
{} \end{cases}
>>

M<<
{} f'(x) = 0; \Rightarrow 3\left(x + 1\right)\left(x - 1\right) = 0; \Rightarrow \\
{} P_1(-1, 2); P_1(1, -2);
>>

M<f'(x) E<gt> 0;> ⇒ M<f> steigt. M<\\>
M<f'(x) E<lt> 0;> ⇒ M<f> fällt.  M<\\>
M<f'(x)   =   0;> ⇒ M<f> hat eine waagrechte Tangente.

=helper MyBook::Helper::Gnuplot

# File automatically generated by Plot.pm

# Global settings
set samples 10000
unset border
set xtics axis
set ytics axis
set xzeroaxis lt -1
set yzeroaxis lt -1

# Coordinate system settings
set title ""
set xlabel "\nx\n"
set ylabel "y\n"
set xrange [ -3.000000 : 3.000000 ]
set yrange [ -4.000000 : 4.000000 ]
set grid
set xtics 1.000000
set ytics 1.000000

# Function definitions
func0(x) = x**3. - 3.*x
func1(x) = 3.*x**2. - 3.

# Plotting
plot func0(x) t "f" w l lt 1, func1(x) t "f'" w l lt 2

=hend