Question

analyze the following graph of ( f(x) ). select all the intervals over which ( f(x) ) is concave down. select all that apply: ( square x < -1 ), ( square -1 < x < 0 ), ( square 0 < x < 1 ), ( square x > 1 )

Explanation:

To determine where \( f(x) \) is concave down, we use the second derivative test: \( f(x) \) is concave down when \( f''(x) < 0 \). Since \( f''(x) \) is the derivative of \( f'(x) \), we analyze the slope of \( f'(x) \):

Step 1: Recall Concavity and \( f''(x) \)

A function \( f(x) \) is concave down when its second derivative \( f''(x) < 0 \). Geometrically, \( f''(x) \) is the slope of the tangent line to \( f'(x) \). So, \( f''(x) < 0 \) when \( f'(x) \) is decreasing (negative slope).

Step 2: Analyze the Graph of \( f'(x) \)

• For \( x < -1 \): The graph of \( f'(x) \) is increasing (slope of \( f'(x) \) is positive), so \( f''(x) > 0 \) (concave up).
• For \( -1 < x < 0 \): The graph of \( f'(x) \) is increasing (slope of \( f'(x) \) is positive), so \( f''(x) > 0 \) (concave up).
• For \( 0 < x < 1 \): The graph of \( f'(x) \) is decreasing (slope of \( f'(x) \) is negative), so \( f''(x) < 0 \) (concave down).
• For \( x > 1 \): The graph of \( f'(x) \) is increasing (slope of \( f'(x) \) is positive), so \( f''(x) > 0 \) (concave up).

Answer:

The interval where \( f(x) \) is concave down is \( 0 < x < 1 \), so we select the option \( \boldsymbol{0 < x < 1} \).