Question

analyze the following graph of ( f ). select all of the intervals over which ( f ) is increasing. select all that apply: ( square x < -2 ), ( square -2 < x < -1 ), ( square -1 < x < 2 ), ( square x > 2 )

Explanation:

Step1: Recall when \( f \) is increasing

A function \( f(x) \) is increasing when its derivative \( f'(x)>0 \). So we need to find intervals where the graph of \( f'(x) \) is above the \( x \)-axis.

Step2: Analyze each interval

• For \( x < -2 \): The graph of \( f'(x) \) is above the \( x \)-axis (positive), so \( f \) is increasing here.
• For \( -2 < x < -1 \): The graph of \( f'(x) \) is still above the \( x \)-axis (positive), so \( f \) is increasing here.
• For \( -1 < x < 2 \): The graph of \( f'(x) \) is below the \( x \)-axis (negative), so \( f \) is decreasing here.
• For \( x > 2 \): The graph of \( f'(x) \) is above the \( x \)-axis (positive), so \( f \) is increasing here.

Answer:

\( \boldsymbol{x < -2} \), \( \boldsymbol{-2 < x < -1} \), \( \boldsymbol{x > 2} \) (the corresponding checkboxes for these intervals should be selected)