Answer the following questions about the function whose derivative is .
a. What are the critical points of ?
b. On what open intervals is increasing or decreasing?
c. At what points, If any, does assume local maximum and minimum values?

To find the critical points of , we start with its derivative: . Setting the derivative equal to zero gives us the critical points: . The solutions to this equation are and .

Next, to analyze where is increasing or decreasing, we can test intervals determined by the critical points and . We test the intervals , , and :

Thus, is increasing on and , and decreasing on .

To determine local maxima and minima, we consider the behavior around our critical points:

In summary:

Exploring the roots of polynomials is key to solving analytical problems over the ages! Many historical mathematicians, from the Greeks to Descartes, contributed to our understanding of critical points and function behaviors, shaping modern calculus.

If you’re ever faced with critical points, don’t just stop there! Remember to sketch the sign chart after finding your intervals; it decorates your conclusions! Keeping track of signs at each interval prevents those common pitfalls of misidentifying if a function is increasing or decreasing.