Practice Tasks - Session 05-07

Function Determination & Funktionsscharen

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Part 1: Quadratic Functions

Problem 1: Three Points (xx)

Find the quadratic function \(f(x) = ax^2 + bx + c\) passing through the points \((0, 3)\), \((1, 6)\), and \((2, 11)\).

Problem 2: Vertex Form (x)

Find the equation of a parabola with vertex at \((-3, 5)\) that passes through the point \((0, -4)\).

Problem 3: Given Maximum (xx)

Find the quadratic function that has a maximum at \((2, 8)\) and passes through the origin.

Problem 4: Point and Slope Conditions (xx)

Find the quadratic function \(f(x) = ax^2 + bx + c\) such that: - \(f(1) = 4\) - \(f'(1) = 3\) - \(f(3) = 6\)

Part 2: Cubic Functions

Problem 5: Four Points (xxx)

Find the cubic function \(f(x) = ax^3 + bx^2 + cx + d\) passing through: \((0, 2)\), \((1, 1)\), \((-1, 3)\), and \((2, 6)\).

Problem 6: Two Extrema (xxxx)

Find the cubic function \(f(x) = ax^3 + bx^2 + cx + d\) that has:

  • A local maximum at \((0, 5)\)
  • A local minimum at \((2, 1)\)

Problem 7: Inflection Point Condition (xxx)

Find the cubic function \(f(x) = ax^3 + bx^2 + cx + d\) such that:

  • \(f(0) = 1\)
  • \(f(1) = 2\)
  • \(f'(0) = 3\)
  • Has an inflection point at \(x = 2\)

Problem 8: Mixed Conditions (xxxx)

Find the cubic function with:

  • Passes through \((1, 4)\)
  • Has a local extremum at \(x = 0\) with \(f(0) = 2\)
  • Has \(f'(2) = 6\)

Part 3: Funktionsscharen (Function Families with Parameters)

Problem 9: Single Parameter (xx)

For the function family \(f_a(x) = ax^2 - 4ax + 5\), find the value of \(a\) such that \(f_a(3) = 2\).

Problem 10: Parameter with Extremum (xxx)

For \(g_t(x) = x^3 - 3tx^2 + 4\), find the value(s) of \(t\) such that \(g_t\) has a local extremum at \(x = 2\).

Problem 11: Two Parameters (xxxx)

Find values of \(a\) and \(b\) such that \(f(x) = x^3 + ax^2 + bx\) has:

  • A local maximum at \(x = 1\) with \(f(1) = 6\)

Problem 12: Parameter with Two Conditions (xxxx)

For the function family \(h_k(x) = kx^3 - 3kx + 2\), find \(k\) such that:

  • \(h_k\) has a local extremum at \(x = 1\)
  • \(h_k(2) = 0\)

Part 4: Business Applications

Problem 13: Cost from Marginal Cost (xx)

A company’s marginal cost function is \(MC(x) = C'(x) = 6x^2 - 8x + 15\), where \(x\) is thousands of units.

The fixed cost (cost when \(x = 0\)) is €2000.

Find the total cost function \(C(x)\).

Problem 14: Revenue Function (xxx)

A company knows the following about its revenue function \(R(x) = ax^3 + bx^2 + cx\) (in thousands):

  • Revenue from selling 1000 units (x=1) is €50,000: \(R(1) = 50\)
  • Revenue from selling 2000 units (x=2) is €140,000: \(R(2) = 140\)
  • Marginal revenue at \(x = 1\) is €70,000 per thousand units: \(R'(1) = 70\)

Find the revenue function.

Problem 15: Profit Optimization Design (xxxx)

An analyst wants to model a company’s profit function as cubic: \(P(x) = ax^3 + bx^2 + cx + d\) where \(x\) is production level in thousands.

Requirements:

  • Fixed costs (losses when nothing is produced): \(P(0) = -20\) (€20,000 loss)
  • Break-even at 2000 units: \(P(2) = 0\)
  • Maximum profit at 4000 units: Critical point at \(x = 4\)
  • Profit at maximum is €60,000: \(P(4) = 60\)

Find the profit function.

Part 5: Advanced Funktionsscharen (Exam Practice)

These problems are typical exam questions. Master them!

Problem 17: Zeros with Discriminant (xxx)

For the function family \(f_t(x) = x^2 + 2tx + t + 6\):

  1. For which values of \(t\) does \(f_t\) have exactly two distinct zeros?

  2. For which values of \(t\) does \(f_t\) have exactly one zero?

  3. For which values of \(t\) does \(f_t\) have no zeros?

Problem 18: Extremum Location (xxx)

For \(g_t(x) = x^3 - 3tx^2 + 12x\):

  1. Find the value of \(t\) such that \(g_t\) has a local extremum at \(x = 2\).

  2. For that value of \(t\), classify the extremum.

  3. Find \(g_t(2)\) for that value of \(t\).

Problem 19: Inflection Point Parameter (xxx)

For \(h_t(x) = x^3 + tx^2 - 9x + 5\):

  1. Find \(t\) such that \(h_t\) has an inflection point at \(x = 1\).

  2. For that \(t\), find the coordinates of the inflection point.

Problem 20: Function Value Condition (xx)

For \(f_k(x) = kx^2 - 4x + k\):

  1. Find \(k\) such that \(f_k(1) = 3\).

  2. For that \(k\), find the vertex of the parabola.

  3. Does \(f_k\) open upward or downward?

Problem 21: Two Zeros at Specific Points (xxxx)

For \(g_t(x) = x^2 - tx + t - 3\):

Find \(t\) such that \(g_t\) has zeros at \(x = 1\) and \(x = 3\).

Problem 22: Maximum Value Parameter (xxx)

For \(p_t(x) = -x^2 + 4x + t\):

  1. What is the maximum value of \(p_t\) (in terms of \(t\))?

  2. Find \(t\) such that the maximum value is 10.

  3. For that \(t\), find the zeros of \(p_t\).