Question

(a) sketch a graph that is always increasing but starts out concave up and then changes to concave down. there should be a point of inflection in your picture.

Explanation:

Step1: Recall function - behavior rules

A function is increasing when its first - derivative $f'(x)>0$. It is concave up when $f''(x)>0$ and concave down when $f''(x)<0$. The point of inflection occurs where $f''(x)$ changes sign.

Step2: Sketch the general shape

Start with a curve that is increasing. In the beginning, draw it as a curve that is bending upwards (concave up). Then, draw a point where the curve changes its bending behavior from upwards to downwards (concave down). This change - of - concavity point is the point of inflection.

Answer:

A correct sketch would start with a curve that is rising and bending upwards (like the left - hand side of a parabola opening upwards), then at a certain point, the curve would start to bend downwards (like the right - hand side of an inverted parabola opening downwards) while still rising overall. The point where the bending changes is the point of inflection. Since this is a sketching task and no specific equations are given, a verbal description of the correct shape is provided. A visual sketch would show a smooth curve with the described characteristics on a coordinate plane with x and y axes labeled.