WiSe 2025/2026 - Mock Exam 05
BFP Mathematics Course

Grading

Problem 1	Problem 2	Problem 3	Total	Grade
___ /36	___ /39	___ /25	___ /100	___

Exam Information

Working Time	180 minutes
Total Points	100

Student Information

Name

________________________________________________

Guidelines

The exam duration is 180 minutes (100 points). All 3 problems must be completed.
Please write your name and student ID number on the cover sheet and each worksheet.
Only use the provided paper. Using other papers will invalidate the exam.
The task sheets are part of the examination and must be submitted.
Pencils and red pens are not allowed.
During the exam, conversations (except quietly with the supervisor), copying from others, and holding up work are considered attempts at cheating.
Only writing materials, a non-programmable, non-graphing calculator, drawing instruments, and a monolingual dictionary may be used.
No formula sheets, notes, or books are permitted.
Carrying smartphones, mobile phones, tablets, smartwatches, and similar devices, even when turned off, is prohibited and considered an attempt at cheating.

I wish you much success!

Problem 1: Service Industry Cost Analysis [36 pts. total]

A consulting firm analyzes its cost structure for providing business advisory services. Financial analysis reveals the following information:

The fixed costs are 180 currency units (abbreviated as “CU” hereafter). At 2 quantity units of service (abbreviated as “Un” hereafter) the marginal costs are 12 CU. At 3 Un the curvature of the cost function changes sign. At 10 Un the total costs amount to 980 CU.

Market research shows that the demand for their services follows a linear function assuming a maximum price of 256 CU and a market saturation at 8 Un.

Part A: Function Development

Determine the cost function \(K(x)\), assuming it is a polynomial of third degree. [13 pts.]

Determine the linear demand function \(p(x)\). [3 pts.]

For verification purposes only:

\(K(x) = 2x^3 - 18x^2 + 60x + 180\)

\(p(x) = -32x + 256\)

Determine the revenue function \(E(x)\) and show that it can be written as \(E(x) = -32x^2 + 256x\). [2 pts.]

Show that the profit function is given by \(G(x) = -2x^3 - 14x^2 + 196x - 180\). [2 pts.]

Part B: Optimization and Business Strategy

Prove that the break-even point is at 1 Un of produce. Explain the significance of this quantity for the company. [3 pts.]

Compute the maximum profit and prove it really is a maximum. [4 pts.]

Decide which price the company has to ask for in order to gain the maximum profit. [3 pts.]

Determine the marginal profit at the profit-maximizing production level. Interpret what this value tells us about the firm’s pricing strategy. [2 pts.]

Compute the minimum variable cost per unit and the short-term lower limit price. [4 pts.]

Problem 2: Exponential Function Analysis [39 pts. total]

Let the function \(f\) be given by \(f(x) = x \cdot e^{-x/2}, \quad x \in \mathbb{R}\)

Part I: Basic Properties and Behavior

Determine the domain of the function \(f\). [1 pt.]

Investigate the asymptotic behavior of \(f\), and determine the \(x\)- and \(y\)-intercepts. [8 pts.]

Explain in complete sentences what the asymptotic behavior tells us about the long-term behavior of this function. [2 pts.]

Part II: Critical Analysis

Compute the first derivative \(f'(x)\) using the product rule. [4 pts.]

Determine the nature (classify as local maximum, local minimum, or saddle point) and the coordinates of all stationary points (critical points where \(f'(x) = 0\)). [5 pts.]

Find the equation of the tangent line at the point \(\left(2, \frac{2}{e}\right)\). [4 pts.]

Compute the angle of intersection \(\alpha\) between the tangent and the \(x\)-axis. [2 pts.]

Find the second derivative \(f''(x)\) and determine where the function changes concavity (inflection points). [4 pts.]

For verification purposes only:

\(f'(x) = e^{-x/2}(1 - x/2)\)

Maximum at \(x = 2\)

Argue whether the following statements are true or false: [3 pts.]
1. \(f''(2) = 0\)
2. In the interval \(0 < x < 2\), the function \(f\) is concave down.
3. \(f'(x)\) possesses a maximum at \(x = 2\).
4. The tangential gradient of \(f\) at \(x = 0\) is bigger than the slope of the secant in the interval \([0, 2]\).

Sketch the graph \(G_f\) in the interval \([-2; 10]\). Label stationary points, inflection points, and intercepts. [6 pts.]

Problem 3: Function Determination and Parameter Analysis [25 pts. total]

Part A: Revenue Function Determination

An e-commerce company’s revenue function is modeled by a cubic polynomial \(R(x) = ax^3 + bx^2 + cx + d\), where \(x\) represents thousands of customers served per month.

The following conditions are established from historical data:

When serving zero customers, revenue is zero: \(R(0) = 0\)
At 4 thousand customers, the revenue function has an inflection point with a value of 256 CU
Revenue reaches a local maximum at 8 thousand customers

Translate each condition into a mathematical equation. Explain why the inflection point condition at \((4, 256)\) provides two equations. [4 pts.]

Set up and solve the complete system of equations to find \(a\), \(b\), \(c\), and \(d\). [8 pts.]

Verify that your function satisfies the conditions \(R(4) = 256\) and \(R''(4) = 0\). [2 pts.]

For verification purposes only:

\(R(x) = -2x^3 + 24x^2\)

Part B: Function Family Investigation

Consider the family of functions defined by: \[f_a(x) = (x - a)^2 \cdot e^{-x}, \quad x \in \mathbb{R}, \quad a \in \mathbb{R}\]

Show that for all values of the parameter \(a\), each function \(f_a\) has exactly one zero. State the coordinates of this zero in terms of \(a\). [3 pts.]

Finding the local maximum:
1. Show that the first derivative can be written as: \[f_a'(x) = e^{-x}(x - a)(a + 2 - x)\] [2 pts.]

Using the factored form above, find all critical points of \(f_a\). Explain why \(x = a\) is not a local extremum. [2 pts.]

Verify that the local maximum occurs at \(x = a + 2\) and calculate \(f_a(a+2)\). [2 pts.]

The maximum value of \(f_a\) is \(y_{max} = 4e^{-(a+2)}\).
1. Calculate the maximum height when \(a = 0\). [1 pt.]

For which value of \(a\) does the maximum height equal \(4e^{-3}\)? [1 pt.]

Grading Reference

Grade	Percentage
1 (Excellent)	≥ 90%
2 (Very Good)	≥ 77%
3 (Good)	≥ 63%
4- (Pass)	≥ 45%
5-6 (Fail)	< 45%

Note: Passing grade requires at least 45% of total points.

Appendix A – Terms used in phrasing of problems

Verb	Task
name, state, give	A reasoning does not have to be given unless explicitly demanded.
decide	A reasoning does not have to be given unless explicitly demanded.
assess	The judgment provided needs to be explained.
describe, characterize	A description requires suitable wording and usage of technical terminology. A reasoning does not have to be provided.
explain, illustrate	The explanation provides information which allows to comprehend a graphical depiction or a mathematical procedure.
interpret, construe	An interpretation establishes a relation between e.g. a graphical depiction, a term or the result of a calculation and the provided context.
substantiate, reason, prove, show	Statements and issues are to be confirmed by logical induction. The method can be freely chosen unless stated otherwise. The chosen method needs to be explained.
evaluate, calculate, compute, verify	The computation needs to be illustrated starting from an ansatz.
determine, identify	The method can be freely chosen unless stated otherwise. The chosen method needs to be explained.
investigate	The method can be freely chosen unless stated otherwise. The chosen method needs to be explained.
graph, plot	All diagrams and plots have to be drawn accurately with care.
sketch	The sketch needs to contain all essential pieces of information.