Problem 8 - Calculus Early Transcendentals

Question

Sketch the graph of a function \(f\) that has a local minimum value at a point \(c\) where \(f^{\prime}(c)\) is undefined.

Step-by-Step Solution

Answer

Answer: The primary characteristic of such a function is that it has a sharp turn, like a cusp or a corner, at the point \((c, f(c))\).

Step 1: Choose a point c for the local minimum

Choose a point on the coordinate plane where the function will have a local minimum value. Let this point be \((c, f(c))\).

Step 2: Sketch a sharp turn at point c

At the point \((c, f(c))\), draw a cusp or corner, ensuring that the value of \(f(x)\) for \(x < c\) is higher than the value at \(c\) (i.e., the graph is decreasing), and that the value of \(f(x)\) for \(x > c\) is also higher than the value at \(c\) (i.e., the graph is increasing). This will create a sharp turn in the graph, causing the derivative \(f^{\prime}(c)\) to be undefined at the point \((c, f(c))\).

Step 3: Connect the graph around point c

Sketch the remainder of the graph to the left and right of the point \((c, f(c))\), making sure that the graph shows a smooth transition from the decreasing region to the increasing region, maintaining the sharp turn at \((c, f(c))\). Since this is not a specific function, the shape of the graph before and after point \((c, f(c))\) is up to interpretation, as long as the overall conditions are met: the function has a local minimum at point \(c\), and its derivative is undefined at that point. By following the above steps, we have sketched a function \(f\) that has a local minimum value at point \(c\), and its derivative is undefined at that point.