Question

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

Explanation:

Step1: Recall function - behavior rules

A decreasing function has a negative - slope ($y'<0$). Concave - down means $y''<0$ and concave - up means $y''>0$. The point of inflection is where $y''$ changes sign.

Step2: Analyze the options

We need a graph that starts with a negative and decreasing slope (concave - down) and then has a negative but increasing slope (concave - up).
The correct graph will have a smooth curve that is always going down (negative slope) and changes its curvature from bending downwards to bending upwards at the point of inflection.

Answer:

The graph that starts out with a steep negative slope, curves downwards initially (concave - down), then has a point where the curvature changes and starts curving upwards (concave - up) while still having a negative slope throughout is the correct one. Without seeing the specific options clearly labeled, in general, look for a curve that is always decreasing (going from top - left to bottom - right) and has a change in the direction of its bend (from down - bending to up - bending).