Exercise 3 | Wiskunde op Tilburg University

Determine all the extrema of

$y(x)=\textrm{ln}(x)-2x$ .

Antwoord 1 correct

Correct

Antwoord 2 optie

Antwoord 2 correct

Fout

Antwoord 3 optie

Antwoord 3 correct

Fout

Antwoord 4 optie

$y(\frac{1}{2})=\textrm{ln}(\frac{1}{2})-1$ is a minimum.

Antwoord 4 correct

Fout

Antwoord 1 optie

$y(\frac{1}{2})=\textrm{ln}(\frac{1}{2})-1$ is a maximum.

Antwoord 1 feedback

Correct:

$y'(x)=\frac{1}{x}-2$ and hence,

$y'(x)=0$ for

$x=\frac{1}{2}$ . Since

$y(\frac{1}{2})=\textrm{ln}(\frac{1}{2})-1$ ,

$y(\frac{1}{4})=\textrm{ln}(\frac{1}{4})-\frac{1}{2}$ and

$y(1)=-2$ it holds that

$y(\frac{1}{2})=\textrm{ln}(\frac{1}{2})-1$ is a maximum.

Go on.

Antwoord 2 feedback

Wrong:

$x=0$ is outside the domain of the function ln

$(x)$ . Hence, there is no boundary extremum.

See Extra explanation: natural logarithm.

Antwoord 3 feedback

Wrong:

$x=0$ is outside the domain of the function ln

$(x)$ . Hence, there is no boundary extremum.

See Extra explanation: natural logarithm.

Antwoord 4 feedback

Wrong: Calculate

$y(\frac{1}{2})$ ,

$y(\frac{1}{4})$ and