Questions: The twice-differentiable function f is shown below on the domain (-9,9). The graph of f has points of inflection at x=-2, x=2, indicated by small green circles. What inferences can be made about the graphs of f, f', and f'' when x=-2?

The twice-differentiable function f is shown below on the domain (-9,9). The graph of f has points of inflection at x=-2, x=2, indicated by small green circles. What inferences can be made about the graphs of f, f', and f'' when x=-2?

Transcript text: The twice-differentiable function $f$ is shown below on the domain $(-9,9)$. The graph of $f$ has points of inflection at $x=-2$, $x=2$, indicated by small green circles. What inferences can be made about the graphs of $f, f^{\prime}$, and $f^{\prime \prime}$ when $x=-2$?

Solution

Solution Steps

Step 1: Analyze the graph of f at x = -2

At x = -2, the graph of _f_ has an inflection point. This means the concavity of _f_ changes at x = -2. Specifically, _f_ is concave up to the left of x = -2 and concave down to the right of x = -2. The graph of _f_ also appears to have a positive y-value at x = -2.

Step 2: Analyze the graph of f' at x = -2

Since _f_ has an inflection point at x = -2, _f'_ has a local maximum or minimum at x = -2. Looking at the graph, the slope of _f_ is decreasing around x = -2. This means that _f'_ has a local maximum at x = -2. Since _f_ is increasing around x = -2, _f'_ is positive at x = -2.

Step 3: Analyze the graph of f'' at x = -2

Since _f_ has an inflection point at x = -2 where the concavity changes from up to down, _f''_ changes from positive to negative at x = -2. Therefore, _f''_ is zero at x = -2.