Maxima, Minima, Mean Value Theorem | Calculus I A Lecture 9 | Nge Kie Seng 20251029

This Calculus I lecture, delivered by Nge Kie Seng, focuses on applications of derivatives, specifically on finding and classifying extreme values of functions. The instructor begins by reviewing the derivative of exponential functions and then introduces the concepts of absolute and local maxima/minima, using graphical intuition and precise definitions. He emphasizes that local extrema cannot occur at endpoints of the domain. The Extreme Value Theorem is stated, guaranteeing the existence of absolute extrema for continuous functions on closed intervals. The lecture then introduces stationary points (where derivative is zero) and singular points (where derivative does not exist), collectively called critical points. A key theorem states that local extrema must occur at critical points, but not all critical points are extrema. The instructor illustrates these concepts with examples, including a piecewise function and a function with a cusp. He also outlines a strategy for finding absolute extrema on a closed interval by evaluating the function at critical points and endpoints. The lecture concludes with an introduction to the Mean Value Theorem, though the transcription cuts off before its full statement and proof.