Question

use the graph of y = f(x) to discuss the graph of y = f(x). organize your conclusions in a table, and sketch a possible graph of y = f(x). x = 2 | local minimum | inflection point 2 < x < ∞ | negative and decreasing | decreasing and concave downward sketch the graph of f(x). choose the correct graph.

Explanation:

Step1: Analyze local minimum

At $x = 2$, $f(x)$ has a local minimum. This means the graph of $f(x)$ changes from decreasing to increasing at $x=2$.

Step2: Analyze for $2 < x<\infty$

For $2 < x<\infty$, $f^{\prime}(x)$ is negative and decreasing, so $f(x)$ is decreasing and concave - downward.

Step3: Evaluate options

Option A has a local minimum around $x = 2$ and is decreasing and concave - downward for $x>2$. Option B does not have a local minimum at $x = 2$. Option C has an increasing part for $x>2$ which is not correct. Option D does not have the correct concavity for $x>2$.

Answer:

A.