Problem 46 - Essential Calculus

Question

Consider a function \(f\) such that \(f^{\prime}\) is decreasing. Sketch graphs of \(f\) for (a) \(f^{\prime}<0\) and (b) \(f^{\prime}>0\).

Step-by-Step Solution

Answer

When \(f^{\prime}<0\), the graph of the function \(f\) is a downward concave curve since the function is decreasing and slope (its derivative) is getting lesser in negatives. While when \(f^{\prime}>0\), the graph of \(f\) is a concave down curve that is rising, as the function is increasing but its derivative is getting lesser positives. Both graphs indicate that the derivative of \(f\) is decreasing.

Step 1: Sketch the graph when \(f^{\prime}<0\)

When \(f^{\prime}<0\), it implies that \(f\) is a decreasing function. Also, since \(f^{\prime}\) is decreasing, the slope of the function \(f\) is also decreasing, which suggests a downward concave curve. Thus, the graph of the function will be going down, and it's a concave down curve.

Step 2: Sketch the graph when \(f^{\prime}>0\)

On the other hand, when \(f^{\prime}>0\), it signifies that \(f\) is an increasing function. Given that \(f^{\prime}\) is decreasing, the slope of the function \(f\) is becoming less steep, which indicates a concave down curve as well. Thus, the graph of the function will be rising, but it is also a concave down curve.