Absolute minima & maxima (entire domain) | AP Calculus AB | Khan Academy

TL;DR

The video explains how to find the absolute extrema of a function using derivatives, and shows that the function has an absolute minimum but no absolute maximum.

Transcript

• [Voiceover] So we have the function g of x is equal to x squared times the natural log of x. And what I wanna do in this video is see if you can figure out the absolute extrema for g of x. So are there x values where g takes on an absolute maximum value, or an absolute minimum value. Sometimes you might call them a global maximum, or a global min... Read More

Key Insights

• 👈 The natural log of x is only defined for x > 0, and its domain must be considered when finding critical points of a function.
• 😥 Critical points are points where the derivative is either zero or undefined.
• 😥 The intervals around the critical point can be analyzed to determine if the function has an absolute minimum or maximum in those intervals.

Questions & Answers

Q: What is the domain of the function g(x)?

The domain of g(x) is all real numbers greater than zero, as the natural log of x is only defined for x > 0.

Q: How is the derivative of the function g(x) calculated?

The derivative g'(x) is calculated using the product rule, resulting in 2xln(x) + x.

Q: What are the critical points of the function g(x)?

The only critical point is x = 1/sqrt(e), where g'(x) is equal to zero.

Q: Does the function g(x) have an absolute maximum?

No, the function does not have an absolute maximum as it continues to increase indefinitely for x > 1/sqrt(e).

Summary & Key Takeaways

• The natural log of x is only defined for x > 0, and the domain of the function is all real numbers greater than zero.

• The derivative of the function is found using the product rule, and the critical points are determined where the derivative is equal to zero or undefined.

• The critical point is found to be x = 1/sqrt(e), and by analyzing the intervals around the critical point, it is determined that there is an absolute minimum but no absolute maximum.

• The graph of the function confirms the results obtained analytically.