Finding the Absolute Extrema on a Closed Interval Example with f(x) = x^3 - 12x

TL;DR

Derive, find critical numbers, plug into function, obtain absolute max/min on closed interval 0,6.

Transcript

find the absolute extrema of this function here x cubed minus 12x on the closed interval zero comma six so solution the first step is to find the critical numbers so find the critical numbers so we'll call them cns that's the first step so you start by taking the derivative so f prime of x let's see so here you have x cubed so when you take that de... Read More

Key Insights

• #️⃣ Critical numbers are where the derivative of a function is zero.
• 😁 To find absolute max/min, evaluate function at critical numbers and interval endpoints.
• 🤯 Absolute max is the highest function value, while the absolute min is the lowest.
• #️⃣ Derivatives are essential in determining critical numbers for optimization.

Questions & Answers

Q: How do you find critical numbers for a function?

Critical numbers are found by setting the derivative of the function to zero. In this case, it is 3x^2 - 12 = 0, leading to x = 2 as the critical number.

Q: When finding absolute extrema, why do we only consider where the derivative is zero?

We only consider where the derivative is zero because critical numbers represent points where the function may have a maximum, minimum, or an inflection point.

Q: Why is -2 not a critical number in this case?

The critical number -2 is disregarded because it is outside the interval [0, 6]. We are only concerned with critical numbers within the defined interval.

Q: How do we identify the absolute maximum and minimum values for a function on a given interval?

To determine absolute max/min, evaluate the function at critical numbers and endpoints within the interval, choosing the highest value as the max and the lowest as the min.

Summary & Key Takeaways

• Derive the function x^3 - 12x to get 3x^2 - 12.

• Find critical numbers by setting the derivative to zero.

• Plug critical numbers and endpoints (0, 6) into the function to find absolute max/min.