Question

question
analyze the following graph of ( f(x) ). over what interval is ( f ) concave down?
give your answer in interval notation.
provide your answer below:

Explanation:

Step1: Recall concavity and derivative rules

A function \( f(x) \) is concave down when its second derivative \( f''(x) < 0 \). Since \( f''(x) \) is the derivative of \( f'(x) \), we need to find where \( f'(x) \) is decreasing (because the derivative of a function being negative means the function is decreasing).

Step2: Analyze the graph of \( f'(x) \)

The graph of \( f'(x) \) is a downward - opening parabola (since it has a maximum at \( x = 0 \)). For a parabola \( y = ax^{2}+bx + c \) with \( a<0 \) (downward - opening), the function is increasing on the interval \( (-\infty, h) \) and decreasing on the interval \( (h,+\infty) \), where \( h \) is the x - coordinate of the vertex. Here, the vertex of the parabola \( f'(x) \) is at \( x = 0 \). So \( f'(x) \) is increasing on \( (-\infty, 0) \) and decreasing on \( (0,+\infty) \).

Since \( f''(x) \) is the derivative of \( f'(x) \), \( f''(x)<0 \) when \( f'(x) \) is decreasing. \( f'(x) \) is decreasing for \( x>0 \).

Answer:

\((0,+\infty)\)