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  2. For business economics
  3. Chapter 5: Optimization
  4. Optimization functions of one variable
  5. Monotonicity condition extremum
  6. Exercise 4

Exercise 4

Determine all the extrema of $f(x)=(x-2)^3$.
There are no extrema.
$f(2)=0$ is a maximum.
$f(2)=0$ is a minimum.
$x=2$ is a maximum.
Determine all the extrema of $f(x)=(x-2)^3$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$f(2)=0$ is a maximum.
Antwoord 2 correct
Fout
Antwoord 3 optie
$f(2)=0$ is a minimum.
Antwoord 3 correct
Fout
Antwoord 4 optie
$x=2$ is a maximum.
Antwoord 4 correct
Fout
Antwoord 1 optie
There are no extrema.
Antwoord 1 feedback
Correct: $f'(x)=3(x-2)^2$. Hence, $f'(x)=0$ if $x=2$. Via a sign chart (with for instance $f'(0)=12$ and $f'(4)=12$) we find that $x=2$ is not an extremum location, but a saddle point. Hence, there are no extrema.

Go on.
Antwoord 2 feedback
Wrong: Not every stationary point is an extremum location.

See First-order condition extremum.
Antwoord 3 feedback
Wrong: Not every stationary point is an extremum location.

See First-order condition extremum.
Antwoord 4 feedback
Wrong: An extremum is never a value of the input variable.

Seee Minimum/maximum.