Question

Sketch the graph of a differentiable function \(f\) such that \(f>0\) and \(f^{\prime}<0\) for all real numbers \(x\).

Short answer

The graph of a function that is always positive and decreasing would start in the first quadrant and approach the X-axis as it moves to the right, never touching or crossing it.

Step-by-step solution

Graph Initialization

The graph initialization is done by marking the X and Y axes. Consider the intersection of X and Y axes as the origin, where \(x = 0\) and \(y = 0\). You can also consider adding tick marks along the axes in order to better visualize the scale of the graph.

Determine Function Behavior

As given, \(f > 0\) and \(f' < 0\) for all real numbers \(x\). This implies that on the graph, all points of the function will be above the X-axis (since \(f > 0\)), and the function shows decreasing behavior (since \(f' < 0\)).

Sketch the Graph

Given that the function has no real roots and is always decreasing, start the sketch from any point in the 1st quadrant (where both X and Y values are positive) and draw a decreasing curve toward the X-axis. However, it's crucial to make sure the curve doesn't touch or cross the X-axis as the function is positive for all real numbers, meaning it never becomes zero or negative.