Class Notes (1,100,000)

CA (630,000)

UTSC (30,000)

Mathematics (1,000)

MATA32H3 (200)

Karimian Pour, C. (10)

Lecture 19

Department

MathematicsCourse Code

MATA32H3Professor

Karimian Pour, C.Lecture

19This

**preview**shows half of the first page. to view the full**2 pages of the document.** MATA32H3-Lec19-2nd Derivative Test and Applied Optimization Problem

2nd Derivative Test

Let y=f(x) be a function such that f’(c)=0

1. If f’(c)>0 local MIN at c

2. If f’(c)<0 local MAX at c

3. If f’(c)=0 or f’(c) is undefined 2nd derivative is of no help

Example:

(when the 2nd derivative is easier than the 1st)

Let y=f(x)=x^2e^-x

Find the critical point using 2nd Derivative Test for maximum/minimum

Solution:

f’(x)=2xe^-x +x^2e^-x (-1) = e^-x (2x-x^2)

= e^-x (2-x) x

The critical points are x=0 and x=2.

f’’(x)= -e^(2x-x^2) + e^-x (2-2x)

= e^-x (-2x+x^2+2-2x)

f’’(x)= e^-x (x^2-4x+2)

f’’(0)= 1 (2) = 2 f has local minimum at 0

f’’(2)= e^-2 (4-8+2) = e^-2 (-2) < 0 f has local maximum at 2

1st Derivative

Test

2nd Derivative Test

Do not need f’’

Always works

Easy…

Evaluate

Sign change

needs to be

checked

Need f’’

Can fail

NA if f’(c) is

undefined

###### You're Reading a Preview

Unlock to view full version

Subscribers Only