Question

five points are labeled on the graph of $y = f(x)$ below. select all the points that fall on intervals where the function is concave up.

Explanation:

Step1: Recall the definition of concave up

A function \(y = f(x)\) is concave up on an interval if the graph of the function lies above its tangent lines on that interval. Geometrically, the second - derivative \(f''(x)>0\) on the interval (if the function is twice - differentiable), and the graph has a "cup - like" shape.

Step2: Analyze the graph

• For points \(a\) and \(b\): The graph of the function \(y = f(x)\) in the region where points \(a\) and \(b\) are located has a "cup - like" shape. The function is concave up in this region.
• For point \(c\): The graph of the function in the region where point \(c\) is located has a "cap - like" shape (concave down). The second - derivative \(f''(x)<0\) in this region.
• For point \(d\): Point \(d\) is an \(x\) - intercept. To the left of \(d\) (near \(c\)), the function is concave down, and to the right of \(d\) (near \(e\)), the function is concave up. But the point \(d\) itself is not in a region of concavity (it is on the \(x\) - axis and is a transition point in terms of the sign of the second - derivative).
• For point \(e\): The graph of the function in the region where point \(e\) is located has a "cup - like" shape. The function is concave up in this region.

Answer:

Points \(a\), \(b\), and \(e\) fall on intervals where the function is concave up.