Question

Group Activity In Exercises \(39-42,\) sketch a graph of a differentiable function \(y=f(x)\) that has the given properties. $$\begin{array}{l}{f^{\prime}(-1)=f^{\prime}(1)=0, \quad f^{\prime}(x)>0 \text { on }(-1,1)} \\ {f^{\prime}(x)<0 \text { for } x<-1, \quad f^{\prime}(x)>0 \text { for } x>1}\end{array}$$

Short answer

The graph sketched would show three parts. The function \(f(x)\) decreasing for \(x<-1\), reaching a minimum at \(x=-1\), then increasing from \(x=-1\) to \(x=1\) to reach a maximum, and after \(x=1\), continue increasing for \(x>1\). The exact shape of the graph cannot be determined as no specific form of the function is given.

Step-by-step solution

Understand the function behavior based on derivative

Analyze the given conditions for \(f'(x)\) which is the derivative of the function. When \(f'(x)>0\), the function \(f(x)\) is increasing. When \(f'(x)<0\), the function \(f(x)\) is decreasing. We are given that \(f'(x)>0\) for \(x\) being in the intervals (-1,1) and \(x>1\), and \(f'(x)<0\) for \(x<-1\). This suggests that the function is increasing on the intervals (-1,1) and \(x>1\), and decreasing for \(x<-1\). At \(x=-1\) and \(x=1\), \(f'(x)=0\), thus these values are either local maximum or minimum.

Identify local extrema

Identify and label the maximum and the minimum points. This can be done by observing the function's behavior before and after these points. Since \(f'(x)\) shifts from negative to positive at \(x=-1\), this point is a local minimum. Since the value of \(f'(x)\) shifts from positive to negative at \(x=1\), this point is a local maximum.

Sketch the graph

Begin by plotting the points \((-1, f(-1))\) and \((1, f(1))\) which are local minimum and maximum respectively. Since details about \(f(x)\) are not given, these points can be plotted anywhere convenient on the \(y\)-axis as far as they reflect the relationship that \(f(-1)1\), the graph should be increasing. Draw it from the point \((1, f(1))\) sloping upwards.