Question

find the numbers, if any, at which f has a local minimum. what are the local minima?

a. f has a local minimum at \\(x = -\pi\\) and \\(\pi\\); the local minimum is 2
b. f has no local minimum

Explanation:

Identify local minimum candidates

Using the Local Extrema and Graph Analysis knowledge points

$$ \text{A local minimum occurs at } x = c \text{ if } f(c) \le f(x) \text{ for all } x \text{ in an open interval containing } c. $$

Looking at the graph, the lowest point in its immediate neighborhood is at the origin's vertical axis intersection:

$$ (x, y) = (0, -2) $$

Evaluate the local minimum value

Using the Local Extrema and Graph Analysis knowledge points

$$
LATEXBLOCK0
$$

Compare with given options

Using the Graph Analysis knowledge point

$$
LATEXBLOCK1
$$

Since the actual local minimum is at \(x = 0\) with value \(-2\), neither Option A nor Option B correctly describes this local minimum. However, looking at the cropped options, the correct option containing \(x = 0\) and \(-2\) is cut off. Among the visible options, Option A is incorrect because \(x = -\pi\) and \(x = \pi\) are local maxima where the value is \(2\). Option B is incorrect because a local minimum does exist at \(x = 0\).

Answer:

• A. f has a local minimum at x = -\pi and \pi; the local minimum is 2
• B. f has no local minimum