Exercise 2 | Wiskunde op Tilburg University

Consider the function

$y(x)=x^4-\frac{3}{2}x^2$ . Which of the following statements is true?

Antwoord 1 correct

Correct

Antwoord 2 optie

The function is convex on the interval

$(-\infty,\frac{1}{2}]$ and concave on the interval

$[\frac{1}{2},\infty)$ .

Antwoord 2 correct

Fout

Antwoord 3 optie

The function is convex on the entire interval.

Antwoord 3 correct

Fout

Antwoord 4 optie

The function is convex on the interval

$(-\infty,0]$ , and concave on the interval

$[0,\infty)$ .

Antwoord 4 correct

Fout

Antwoord 1 optie

The function is convex on the intervals

$(-\infty,-\frac{1}{2}]$ and

$[\frac{1}{2},\infty)$ , and concave on the interval

$[-\frac{1}{2},\frac{1}{2}]$ .

Antwoord 1 feedback

Correct: From

$y''(x)=12x^2-3$ it follows that

$y''(x)\geq0$ for every

$x\leq-\frac{1}{2}$ and

$x\geq\frac{1}{2}$ , and that

$y''(x)\leq0$ for all other values of

$x$ .

Go on.

Antwoord 2 feedback

Wrong: Note that the point

$x=-\frac{1}{2}$ is also an inflection point.

See inflection point.

Antwoord 3 feedback

Wrong: The function has two inflection points.

See inflection point.

Antwoord 4 feedback

Wrong:

$x=0$ is not an inflection point.

See inflection point.