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  2. For business economics
  3. Chapter 5: Optimization
  4. Optimization convex/concave functions
  5. Functions of one variable
  6. Second-order condition
  7. Exercise 2

Exercise 2

Consider the function $y(x)=3x^2+e^{2x}$ on the interval $x<0$. Which of the following statements is true?
The function is convex
The function is concave.
The functions is neither convex nor concave.
None of the others statements are correct.
Consider the function $y(x)=3x^2+e^{2x}$ on the interval $x<0$. Which of the following statements is true?
Antwoord 1 correct
Correct
Antwoord 2 optie
The function is concave.
Antwoord 2 correct
Fout
Antwoord 3 optie
The functions is neither convex nor concave.
Antwoord 3 correct
Fout
Antwoord 4 optie
None of the others statements are correct.
Antwoord 4 correct
Fout
Antwoord 1 optie
The function is convex
Antwoord 1 feedback
Correct: From $y''(x)=6+4e^{2x}$ it follows that $y''(x)>0$ for every $x<0$. Hence, the function is convex.

Go on.
Antwoord 2 feedback
Wrong: Note that $e^{2x}>0$ for every $x$, hence also for $x<0$.

Try again.
Antwoord 3 feedback
Wrong: Determine the second-order derivative $y''(x)$ and use the second-order condition.

See second-order condition.
Antwoord 4 feedback
Wrong: Determine the second-order derivative $y''(x)$ and use the second-order condition.

See second-order condition.