Mathematical tools for intermediate economics classes
Iftekher Hossain
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Derivatives and their uses

Section 6

Practise questions

Question 1

For each of the following functions check whether the function is concave or convex around the neighbourhood of the given point.

  1. \(f(x) = 30x - x^{2}\) at \(x = 15\)
  2. \(f(x) = -x^{3} - 6x^{2} + 1440x - 600\) at \(x = 20\)
  3. \(f(x) = 3x^{3} - 12x + 50\) at \(x =2\)

Hint: Evaluate the second-order derivatives at the given points and check the sign.

  1. Concave
  2. Concave
  3. Convex

Question 2

For each of the following functions find the critical points (find the points at which \({f}'(x) = 0\)).

  1. \(f(x) = 30x - x^{2}\)
  2. \(f(x) = -x^{2} + 12x - 24\)
  3. \(f(x) = 3x^{2} - 12x + 50\)
  4. \(f(x) = -x^{3} - 6x^{2} + 1440x - 600\)
  5. \(f(x) = x^{2} + 2x\)

Suggestion: Set the first-order derivative equals zero and solve the equation.

  1. \(x = 15\)
  2. \(x = 6\)
  3. \(x = 2\)
  4. \(x = 20\), \(x = -24\)
  5. \(x = -1\)

Question 3

Using the properties of derivatives, draw the following functions.

  1. \(f(x) = 20x - x^{2}\)
  2. \(f(x) = 2x^{2} - 12x + 20\)
  3. \(f(x) = 3x^{2} - 36x + 750\)
  4. \(f(x) = x^{3} - 6x^{2} + 30x + 750\)

Suggestion: Set the first-order derivative equals zero and solve the equation.

  1. Critical value \(x = 10\), relative max. The graph of the function is similar like the graph in example 2 (Section 6).
  2. Critical value \(x = 3\), relative min. The graph of the function is similar like the graph in example 3 (Section 6).
  3. Critical value \(x = 6\), relative min. The graph of the function is similar like the graph in example 3 (Section 6).
  4. No critical value. Inflection point at \(x = 2\). The graph of the function is similar like the graph in example 4 (Section 6).


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