Question

question analyze the following graph of f(x). select all of the intervals over which the function is concave up. select all that apply. x ≤ -1, -1 < x < 0, 0 < x < 1, x ≥ 1

Explanation:

To determine where the function \( f(x) \) is concave up, we analyze the shape of the graph:

• A function is concave up when its graph curves upward (like a cup).
• For the interval \( -1 < x < 0 \): The graph curves upward in this region.
• For the interval \( x \geq 1 \): The graph also curves upward here.
• For \( x \leq -1 \): The graph curves downward (concave down).
• For \( 0 < x < 1 \): The graph curves downward (concave down).

Step 1: Analyze \( x \leq -1 \)

The graph is curving downward, so not concave up.

Step 2: Analyze \( -1 < x < 0 \)

The graph curves upward, so concave up here.

Step 3: Analyze \( 0 < x < 1 \)

The graph curves downward, so not concave up.

Step 4: Analyze \( x \geq 1 \)

The graph curves upward, so concave up here.

Answer:

• \( -1 < x < 0 \)
• \( x \geq 1 \)