#### Chapter 7 Differential Calculus

Section 7.5 Applications

# 7.5.3 Exercises

With the following exercise the elements of the curve analysis method can be trained:
##### Exercise 7.5.1
Führen Sie für die Funktion $f\left(x\right)=-{x}^{3}-6{x}^{2}-7$ eine vollständige Kurvendiskussion durch.

##### Exercise 7.5.51
Carry out a complete curve analysis for the function $f$ with $f\left(x\right)=\left(2x-{x}^{2}\right)e{}^{x}$ and enter your results into the input fields.
Maximum domain:
(as an interval (a;b)) .

Set of intersection points with the $x$-axis (zeros of $f\left(x\right)$):
(as a set $\left\{$a;b;c$\right\}$, only $x$-components) .

The $y$-intercept is at $y$$=$ .

Symmetry: The function is
 axially symmetric with respect to the $y$-axis, centrally symmetric with respect to the origin.

Limiting behaviour: For $x\to \infty$, the functions values $f\left(x\right)$ tend to
, and for $x\to -\infty$, they tend to .

Derivatives: We have $f\text{'}\left(x\right)$ =
and $f\text{'}\text{'}\left(x\right)$ =
.

Monotony behaviour: The function is monotonically increasing on the interval
and monotonically decreasing otherwise.

Extremal values: The point ${x}_{1}$ =
is a minimum point and the point ${x}_{2}$ =
is a maximum point.

Inflexion points: The set of inflexion points consists of

(as a set, roots can be entered) .

Sketch the graph and compare your result to the sample solution.