Problem 43

Given information

We have been given the function

Step-by-step explanation

To find the first and second derivatives we apply the power rule to each summand: . When we plot both the original function and its derivatives we can see that our derivations and are reasonable. The original function (blue) is always decreasing, so we would expect the slopes of the tangent lines (red) would always be negative, and they are. However, the blue line is decreasing at a decreasing rate, that is, it's decreasing by fewer and fewer amounts as time goes on, this means the slope is getting closer and closer to zero at a slower pace. Thus, when we look at the second derivative (gold) which describes the slope of the tangent lines to the red line it's positive (the red line is increasing towards zero) but decreasing.