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 the function lies below its tangent lines).

Step2: Analyze each interval

• Interval \(A - B\):
• The function is increasing (\(f^{\prime}(x)>0\)).
• To check concavity, we can use the fact that if the graph of \(y = f(x)\) has a "hill - like" shape (the rate of increase is decreasing), it is concave down. In the interval \(A - B\), the function is increasing and the slope of the tangent line is decreasing (the graph is concave down).
• 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 (\(f^{\prime}(x)>0\)).
• But the graph is concave up (the rate of increase is increasing, \(f^{\prime\prime}(x)>0\)).

Answer:

A. the interval from \(A\) to \(B\)