Determine the intervals where the function is decreasing.

Question

Determine the intervals where the function  is decreasing. ​

Quizplus · Accepted Answer

To determine the intervals where this function is decreasing, we need to find where the first derivative is negative. Taking the derivative of the function gives us e^x - 2. Setting this equal to 0 and solving for x gives us x = ln(2). This critical point divides the real number line into two intervals: (-infinity, ln(2)) and (ln(2), infinity). We can check a point in each interval to see which one makes the derivative negative. For example, let's evaluate the derivative at x = 0 (in the interval (-infinity, ln(2))). We get e^0 - 2, which is negative. Therefore, the function is decreasing on this interval. Similarly, if we evaluate the derivative at x = 1 (in the interval (ln(2), infinity)), we get e^1 - 2, which is positive. Therefore, the function is increasing on this interval. The only choice that includes the interval where the function is decreasing is B.

Determine the Intervals Where the Function Is Decreasing