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 second derivative \(f''(x)>0\) on that interval. Geometrically, the graph of the function lies above its tangent lines on intervals where it is concave up. Another way to think about it is that the slope of the tangent line is increasing.

Step2: Analyze the graph

• For a concave - up function, the graph has a “cup - like” shape.
• Looking at the points \(a\) and \(b\): The part of the graph near points \(a\) and \(b\) has a shape where the slope of the tangent line is increasing.
• For point \(c\): The graph near \(c\) is concave down (the slope of the tangent line is decreasing).
• For point \(d\): \(d\) is an \(x\) - intercept. The concavity is determined by the shape of the graph around the point. The graph near \(d\) is concave down (since the function is changing from positive to negative and the slope of the tangent line is decreasing in the neighborhood of \(d\)).
• For point \(e\): The graph near \(e\) is concave up.

Answer:

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