Rog_Sec_14_7_P_2

# Rog_Sec_14_7_P_2 - Math 20C Lecture Examples Section 14.7...

This preview shows pages 1–2. Sign up to view the full content.

(8/20/08) Math 20C. Lecture Examples. Section 14.7, Part 2. Maxima and minima: The second-derivative test With the Second-Derivative Test for local maxima and minima of functions y = f ( x ) of one variable, we find the points a where the first derivative y prime ( x ) is zero and the tangent line to the graph is horizontal. If f primeprime ( a ) is positive, then the graph of f is concave up at x = a and is above the tangent line for x near a , so the function has a local minimum at that point (Figure 1). If f primeprime ( a ) is negative, then the graph of f is concave down at x = a and below the tangent line for x near x = a and the function has a local maximum there (Figure 2). a y = f ( x ) Tangent line x a y = f ( x ) Tangent line x f prime ( a ) = 0 , f primeprime ( a ) > 0 f prime ( a ) = 0 , f primeprime ( a ) < 0 Local minimum Local maximum FIGURE 1 FIGURE 2 To extend this result to two variables, we need to reinterpret it in terms of Taylor polynomials. Recall that the second-degree Taylor polynomial approximation of y = f ( x ) at x = a is T 2 ( x ) = f ( a ) + f prime ( a )( x - a ) + 1 2 f primeprime ( a )( x - a ) 2 . This polynomial has the same value and first two derivatives as y = f ( x ) at a and, consequently, is the second-degree polynomial that best approximates y = f ( x ) near a . If a is a critical point of y = f ( x ), then f prime ( a ) = 0 and the Taylor polynomial is T 2 ( x ) = f ( a ) + 1 2 f primeprime ( a )( x - a ) 2 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern