Question

the figure shows the graph of a function ( f ). the zero and extrema for ( f ) are labeled, and the point of inflection of the graph of ( f ) is labeled. let ( a, b, c, d ), and ( e ) represent the ( x )-coordinates at those points. of the following, on which interval is ( f ) increasing and the graph of ( f ) concave down? (a) the interval from ( a ) to ( b ) (b) the interval from ( b ) to ( c ) (c) the interval from ( c ) to ( d ) (d) the interval from ( d ) to ( e )

Explanation:

Step1: Recall the definitions

• A function \(y = f(x)\) is increasing when \(f^{\prime}(x)>0\) (the slope of the tangent line is positive).
• A function \(y = f(x)\) is concave - down when \(f^{\prime\prime}(x)<0\) (the graph of \(y = f^{\prime}(x)\) is decreasing).

Step2: Analyze each interval

• Interval \(A - B\):
• The function is increasing (\(f^{\prime}(x)>0\)).
• The graph of the function is concave - up (since the slope of the tangent line is increasing, \(f^{\prime\prime}(x)>0\)).
• Interval \(B - C\):
• The function is decreasing (\(f^{\prime}(x)<0\)).
• Interval \(C - D\):
• The function is decreasing (\(f^{\prime}(x)<0\)).
• Interval \(D - E\):
• The function is increasing (\(f^{\prime}(x)>0\)).
• The graph of the function is concave - down (since the slope of the tangent line is decreasing, \(f^{\prime\prime}(x)<0\)).

Answer:

D. the interval from \(D\) to \(E\)