Question

Group Activity In Exercises \(39-42,\) sketch a graph of a differentiable function \(y=f(x)\) that has the given properties. $$\begin{array}{l}{\text { A local minimum value that is greater than one of its local maxi- }} \\ {\text { mum values. }}\end{array}$$

Short answer

The graph of the function \(f(x)\) which fulfills the conditions is the piecewise function where \(f(x) = -x^2\) for \(x < 0\) and \(f(x) = x^2\) for \(x > 0\).

Step-by-step solution

Understand local minimum and maximum

A local minimum is a point in the function where the value of the variable is lower than or equal to the values at nearby points. Conversely, a local maximum is a point where the variable is larger than or equal to values at nearby points.

Sketch function with a local maximum

First, sketch a function with a local maximum. This could be simple, for example \(f(x) = -x^2\). The peak of this parabolic function happens at \(x=0\) and is a local maximum.

Sketch function with a higher local minimum

Now, we must sketch a function with a local minimum that is higher than the previous maximum. Consider a piecewise function merged with the previously drawn function. A suitable function could be \(f(x) = x^2\) for \(x > 0\). This function has a local minimum at \(x=0\) and given that we define it for \(x > 0\), it is always larger than the local maximum of the other function.

Merge and sketch the final graph

Now, merge these two functions into a piecewise function. Define \(f(x) = -x^2\) for \(x < 0\) and \(f(x) = x^2\) for \(x > 0\). This function is still differentiable and fulfills our conditions.