given that f(x) is continuous on (-∞,∞), use the information below to sketch the graph of f. f(x) + + + 0…
given that f(x) is continuous on (-∞,∞), use the information below to sketch the graph of f. f(x) + + + 0 - - - nd + + + + + + 0 + + + x - 2 - 1 1 x - 3 - 2 - 1 1 3 f(x) 1 2 1 2 3 choose the correct graph below. o a. o b. o c. o d.
Answer
Explanation:
Step1: Analyze sign of $f'(x)$
When $f'(x)>0$, $f(x)$ is increasing. When $f'(x) < 0$, $f(x)$ is decreasing. $f'(x)$ is positive on $(-\infty,- 2)\cup(1,\infty)$ and negative on $(-2,1)$. So $f(x)$ is increasing on $(-\infty,-2)$ and $(1,\infty)$, and decreasing on $(-2,1)$.
Step2: Use given points
We have the points $(-3,1),(-2,2),(-1,1),(1,2),(3,3)$ from the table.
Step3: Analyze critical - points
$x = - 2$ and $x = 1$ are critical points. At $x=-2$, $f(x)$ has a local maximum (since $f'(x)$ changes sign from positive to negative). At $x = 1$, $f(x)$ has a local minimum (since $f'(x)$ changes sign from negative to positive).
Answer:
D.