Question

Connecting \(f\) and \(\boldsymbol{f}^{\prime}\) Sketch a smooth curve \(y=f(x)\) through the origin with the properties that \(f^{\prime}(x)<0\) for \(x<0\) and \(f^{\prime}(x)>0\) for \(x>0\)

Short answer

The curve representing \(f(x)\) passes through the origin and decreases for \(x<0\) while increasing for \(x>0\). This shows the relationship between the function and its derivative, in that the negative slope corresponds to a negative rate of change (or derivative), and a positive slope corresponds to a positive rate of change.

Step-by-step solution

Understand the Properties of Derivatives

Given that the derivative of a function represents the function's rate of change, when the derivative of a function is negative for \(x<0\), this means that for all negative \(x\) values, the function is decreasing. Similarly, if the derivative of a function is positive for \(x>0\), this means the function is increasing for all positive \(x\) values.

Sketch the Curve

Based on these properties, sketch a curve that passes through the origin. The curve should decrease for \(x<0\), indicating a negative slope or rate of change, and increase for \(x>0\), indicating a positive slope or rate of change. This curve will represent the function \(y=f(x)\).

Interpret the Curve

The resulting curve should illustrate the properties of the derivatives described in the problem. For \(x<0\), the curve decreases, showing the rate of change is negative. For \(x>0\), the curve increases, showing the rate of change is positive. This graphical representation evidences the connection between \(f(x)\) and \(f^{\prime}(x)\).