=for timestamp
Mi Mär  9 16:05:15 CET 2005

=head3 46. Hausaufgabe

=head4 Selbstgestellte Aufgabe

M<f(x) = x^4 - 4x^2 = x^2\left(x^2 - 4\right) = x^2\left(x + 2\right)\left(x - 2\right);>

=over

=item Nullstellen

M<N_1(-2, 0); \quad N_2(0, 0); \quad N_3(2, 0);>

=item Symmetrie

M<f(-x) = x^4 - 4x^2 = f(x);> ⇒ Symmetrie zur M<y>-Achse

=item Extrema

M<f'(x) = 4x^3 - 8x = 4x\left(x^2 - 2\right) = 4x\left(x +
\sqrt{2}\right)\left(x - \sqrt{2}\right) = 0; \\>
M<f''(x) = 12x^2 - 8;> ⇒

=over

=item M<x_1 = -\sqrt{2};>

M<f''(x_1) = f''(-\sqrt{2}) E<gt> 0; \Rightarrow P_{\mathrm{TIP}}(-\sqrt{2}, -4);>

=item M<x_2 = 0;>

M<f''(x_2) = f''(0) E<lt> 0; \Rightarrow P_{\mathrm{HOP}}(0, 0);>

=item M<x_3 = \sqrt{2};>

M<f''(x_3) = f''(\sqrt{2}) E<gt> 0; \Rightarrow P_{\mathrm{TIP}}(\sqrt{2}, -4);>

=back

=item Monotoniebereiche

M<f'(x) = 4x\left(x + \sqrt{2}\right)\left(x - \sqrt{2}\right) E<gt> 0;> ⇒

M<f> ist in sms in M<\left] -\sqrt{2}, 0 \right[ \cup \left] \sqrt{2}, \infty \right[;>

M<f> ist in smf in M<\left] -\infty, -\sqrt{2} \right[ \cup \left] 0, \sqrt{2} \right[;>

=item Graph

=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 [ -2.500000 : 2.500000 ]
set yrange [ -9.000000 : 9.000000 ]
set grid
set xtics 1.000000
set ytics 2.000000

# Function definitions
func0(x) = x**4. - 4.*x**2.
func1(x) = 4.*x**3. - 8.*x

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

=hend

=back