level 3: fill in the table.\n| $f(x)=x$ | $f(x)=-x^{2}$ | $(-\\infty,0)$ | none | $(0,0)$ |\n| $f(x)=x^{2}$…
level 3: fill in the table.\n| $f(x)=x$ | $f(x)=-x^{2}$ | $(-\\infty,0)$ | none | $(0,0)$ |\n| $f(x)=x^{2}$ | $(-\\infty,\\infty)$ | $f(x)=-x$ | $(0,\\infty)$ |\n| function | graph | concave up | concave down | inflection point |
Answer
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 downwards | None | $(-\infty,\infty)$ | None |
| $f(x)=x^{2}$ | 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 |