MATH 111 Calculus I \(\quad\;\;\) Instructor: Difeng Cai

4.1 Maximum and Minimum Values

Fermat's Theorem

Fermat's Theorem: If \(f\) has local max or min at \(c\), and if \(f'(c)\) exists, then \(f'(c)=0.\)

The converse is not true. ATTENTION: \(f'(c)=0\) does not mean \(f(c)\) is a local max or min.
Counter example. \(f(x)=x^3\). \(f'(0)=0\) but \(f\) has no extrema (local or global).

Example of Fermat's Theorem. \(f(x)=\sin x\) has a local max at \(x=\frac{\pi}{2}\), so \(f'(\frac{\pi}{2})=0\).

Critical Numbers (or Critical Points)

A critical number of a function is a number in the domain of \(f\) such that either \(f'(c)=0\) or \(f'(c)\) does not exist.

Example 1. Find the critical numbers of \(f(x)=6x^3-9x^2-36x.\)
Sol: Solving \(f'(x)=0\) yields \(x=-1\) and \(x=2\). So \(-1, 2\) are critical numbers.

Finding Local Extrema using The First Derivative Test

See the example here

\(f''\) and the concavity of \(f\)

concave upward (CU) and concave downward (CD)

If \(f''(c)=0\), then \(c\) is called an inflection point

The Second Derivative Test

Example. \(f''\) is continuous near \(c\). \(f'(1)=0\) and \(f''(1)=-1\). What can you say about \(f\) at \(x=1\) ?
Sol: \(f(1)\) is a local maximum.

Ex. Discuss the curve \(y=x^3\) inflection point