Figure

Using the First Derivative Test to Find Local Extrema

Find the local extrema of \(f(x)=x^{4}-4x^{3}\).

Solution Since \(f\) is a polynomial function, \(f\) is continuous and differentiable at every real number. We begin by finding the critical numbers of \(f\). \[ f^\prime (x) =4x^{3}-12x^{2}=4x^{2}({x-3}) \]

Figure 33 \(f(x)= x^4-4x^3\)

The critical numbers are \(0\) and \(3\). We use the critical numbers \(0\) and \(3\) to form three intervals, as shown in Figure 32. Then we determine where \(f\) is increasing and where it is decreasing by determining the sign of \(f^\prime (x)\) in each interval. See Table 3.

TABLE 3

Interval	Sign of \({x^{2}}\)	Sign of \({x-3}\)	Sign of \({f^\prime (x) =4x^{2}(x-3)}\)	Conclusion
\((-\infty ,0)\)	Positive \((+)\)	Negative \((-)\)	Negative \((-)\)	\(f\) is decreasing
\((0,3)\)	Positive \((+)\)	Negative \((-)\)	Negative \((-)\)	\(f\) is decreasing
\((3,\infty)\)	Positive \((+)\)	Positive \((+)\)	Positive \((+)\)	\(f\) is increasing

Using the First Derivative Test, \(f\) has neither a local maximum nor a local minimum at \(0\), and \(f\) has a local minimum at \(3\). The local minimum value is \(f(3) =-27\).