f' > 0 and f '' < 0
Work Step by Step
f' > 0: Because the function is increasing, f' > 0. We know that the function is increasing and therefore the slope is positive because there exists a higher x value at x=0 than at x = 2. f'' < 0: Because the function is increasing at a decreasing rate, we know that f '' < 0. If one were to look at the tangent line at x = 0 and x = 2, and compare the slopes, one would find that the slopes are decreasing, meaning f'' < 0. One can also look at the curvature of the graph, and see that the shape looks like an arch. Because it looks like an arch, it is concave down, and, as chapter 3.4 describes, f '' < 0 for a concave down curve.