Question

level 3: fill in the table.

$f(x)=x$	$f(x)=-x^{2}$	$(-\infty,0)$	none	$(0,0)$
$f(x)=x^{2}$	$(-\infty,\infty)$	$f(x)=-x$	$(0,\infty)$
function	graph	concave up	concave down	inflection point

Explanation:

Step1: Analyze $f(x)=-x^{2}$

The second - derivative $f''(x)=-2<0$ for all $x$. So it is concave down on $(-\infty,\infty)$ and has no concave - up interval and no inflection point.

Step2: Analyze $f(x)=x^{2}$

The second - derivative $f''(x) = 2>0$ for all $x$. So it is concave up on $(-\infty,\infty)$ and has no concave - down interval and no inflection point.

Step3: Analyze $f(x)=-x$

The first - derivative $f'(x)=-1$ and the second - derivative $f''(x)=0$. It is a straight line, so it is neither concave up nor concave down and has no inflection point.

Step4: Analyze $f(x)=x$

The first - derivative $f'(x)=1$ and the second - derivative $f''(x)=0$. It is a straight line, so it is neither concave up nor concave down and has no inflection point.

Answer:

Function	Graph	Concave up	Concave down	Inflection point
$f(x)=x^{2}$	Parabola opening upwards	$(-\infty,\infty)$	None	None
$f(x)=-x$	Straight line with negative slope	None	None	None
$f(x)=x$	Straight line with positive slope	None	None	None