Find the Intervals where the Function is Increasing, Decreasing and The Relative Extrema

TL;DR

Learn how to find increasing and decreasing intervals and relative extrema in a polynomial function by finding critical numbers and using the first derivative test.

Transcript

in this problem we have to find the intervals where the function is increasing and decreasing and we also have to find the relative extrema let's go ahead and work through this so the first step in these problems is to find the critical numbers so critical numbers are numbers in the domain of the function where the derivative is 0 or undefined so t... Read More

Key Insights

• 😥 Critical numbers are points where the derivative is zero or undefined.
• 🤘 The first derivative test helps determine if a function is increasing or decreasing based on the sign of the derivative.
• 🫥 A number line and test points are used to find intervals of increasing and decreasing.
• #️⃣ Relative extrema are identified by observing the behavior of the function around critical numbers.
• #️⃣ The y-value of a relative maximum or minimum can be found by plugging the critical number into the original function.
• 🏆 The first derivative test is an intuitive method for identifying relative extrema.
• ❓ Parentheses are used to indicate intervals of increasing or decreasing.

Questions & Answers

Q: What are critical numbers?

Critical numbers are points in the function's domain where the derivative equals zero or is undefined. They help identify potential points of interest, such as relative extrema.

Q: How do you determine if a function is increasing or decreasing using the first derivative?

By plugging test points into the derivative and observing the sign of the resulting values. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.

Q: What is the first derivative test?

The first derivative test is a method used to determine relative extrema in a function. It involves analyzing the sign changes in the derivative around critical numbers to identify maximum or minimum points.

Q: How do you find the y-value of a relative maximum or minimum?

To find the y-value of a relative maximum or minimum, plugging the critical number into the original function will give the corresponding output value.

Summary & Key Takeaways

• The first step in finding increasing and decreasing intervals and relative extrema is to find the critical numbers, which are the points in the function's domain where the derivative equals zero or is undefined.

• Once the critical numbers are found, a number line is drawn and critical numbers are plotted on it. Test points are chosen, and their corresponding values in the derivative are calculated to determine if the function is increasing or decreasing.

• Based on the results, the intervals of increasing and decreasing can be determined, as well as the existence of relative extrema.