Final answer:
To determine where a function changes from increasing to decreasing, you should check for critical points, maximum points, and minimum points. Critical points are where the function's first derivative is zero or undefined; maximum points are peaks, and minimum points are troughs in the function's graph.
Step-by-step explanation:
When looking for when a function changes from increasing to decreasing, there are specific values one should check. These values are known as critical points, maximum points, and minimum points.
Critical points occur when the first derivative of the function is zero or undefined. These points are possible locations where the function can change from increasing to decreasing or vice versa. To determine if it is actually the case, one would need to analyze the first derivative around those points.
Maximum points are where the function reaches a peak; the function increases before the maximum point and decreases after. Checking the points before and after the maximum can help establish the change in behavior. Similarly, minimum points are where the function reaches its lowest value in a certain interval, increasing after the point.
Inflection points, which are points where the second derivative of the function is zero or undefined, do not necessarily indicate a change from increasing to decreasing. However, they might indicate a change in the concavity of the function, which can help in understanding the function's overall behavior.