Question

Sketch the graph of a differentiable function \(f\) such that \(f(2)=0, \quad f^{\prime}<0\) for \(-\infty0\) for \(2

Short answer

The drawn graph of the function should decrease from left to right as it approaches the point (2,0), pass through the point (2,0), and then increase from left to right as it moves away from the point (2, 0). The exact shape of the curve can vary as several different functions could meet the given criteria.

Step-by-step solution

Step 1. Understand the provided information

The information given indicates that the function \(f(x)\) is decreasing for all \(x<2\) and increasing for all \(x>2\). This is derived from the constraints \(f^{\prime}<0\) and \(f^{\prime}>0\). These tell us that the slope of \(f(x)\) is negative before \(x=2\) (indicating a decrease) and positive after \(x=2\) (indicating an increase). Moreover, \(f(2)=0\) tells us that the function goes through the point (2, 0).

Step 2. Plot the changing points and intersection points

Start the sketch by indicating the point (2, 0) on the graph as this is a given intersection point. Moreover, mark \(x=2\) as the point where the function changes from decreasing to increasing.

Step 3. Draw the decreasing and increasing parts

Draw a curve from the left of \(x=2\) towards (2, 0) that decreases (going downwards) as it approaches (2, 0) from the left. After (2, 0), draw a curve that increases (going upwards) as it moves away from (2, 0) to the right.

Step 4. Finalizing the drawing

Your sketch should now show a function that smoothly transitions from decreasing to increasing at \(x=2\), and crosses the x-axis at (2, 0). Re-check the function's conditions and requirements against your sketch to ensure it matches closely. Remember we're not aiming for exact precision but rather to capture the behaviour of the function. The function is not specified uniquely, so multiple sketches could correctly capture these features.