Sketch the graph of a function $f(x)$ which has all of the following properties: 1. $\lim_{x \to -2^-} f(x) = -\infty$ 2. $\lim_{x \to -2^+} f(x) = \infty$ 3. $\lim_{x \to \infty} f(x) = 0$ 4. $f(-2) = 2$ 5. $f(3) = 1$ 6. $f(0) = 0$ 7. $f'(x) > 0$ if $x < -2$ or $x > 3$ 8. $f'(x) < 0$ if $-2 < x < 2$ or $2 < x < 3$ 9. $f'(2) = 0$ 10. $f'(-2) = 0$ 11. $f''(x) > 0$ if $x < -3$ or $x > 2$ 12. $f''(x) < 0$ if $-3 < x < 2$
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Properties 1 and 2: $\lim_{x \to -2^-} f(x) = -\infty$ and $\lim_{x \to -2^+} f(x) = \infty$. These two properties indicate that there is a vertical asymptote at $x = -2$. As $x$ approaches $-2$ from the left, $f(x)$ goes to $-\infty$. As $x$ approaches $-2$ from Show more…
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