# Determining Extrema for f(x) = x^2 - 8x + 9

• scorpa
In summary, Daniel is seeking clarification on finding the extreme values of a function in a given interval. He has checked for stationary points and found one at x = 4, where f(4) = -7. He then questions whether this point can be used to determine the absolute minimum. The person responding suggests using the endpoints of the interval in addition to x = 4, and mentions that the function is continuous on the whole domain and is a parabola. They also mention that the sign of "a" (which is 1 in this case) determines the nature of the extremum, and in this case, it is a minimum.
scorpa
Hello!

I am doing a question about curve sketching and determining extrema and just need some clarification. Here is the question:

Determine the extreme values of the function f(x) = x^2 - 8x + 9, where
-1 is greater than or equal to x is less than or equal to 5.

This is what I have done so far:

I decided to check for stationary points, as extreme values may occur at f'(0)=0.

f'(x) = 2x-8
2x-8 = 0
x = 4

f(4) = 4^2 - 8(4) +9 = -7
(4,-7) may be a max, min or neither.

f(3.9) = (3.9)^2 - 8(3.9)+9 = -6.99
f(4.1) = (4.1)^2 - 8(4.1)+9 = -6.99

Does this mean that this point cannot be used in determining the absolute minimum? I am totally lost at this point. I was going to use the endpoints given to figure out the maximum, but I'm not sure if that is an acceptable method. Thanks for any help you can give me.

You can use the endpoints of the interval as well as x = 4. You are allowed to use the endpoints because they are included in the interval.

The function is continuous on the whole domain.It's a parabola.The theory says that the sign of "a" (=1,in this case) decides the nature of the extremum...In this case,it's a minimum...

That's all there's to it.

Daniel.

## 1. What is the formula for determining extrema?

The formula for determining extrema for a function is to take the first derivative of the function and set it equal to zero.

## 2. How do you find the critical points for a function?

The critical points are the x-values where the first derivative is equal to zero or undefined. To find the critical points, set the first derivative equal to zero and solve for x.

## 3. What is the difference between a local and global extremum?

A local extremum is a high or low point on a function that is only valid within a specific interval. A global extremum is the highest or lowest point on a function over its entire domain.

## 4. How can you determine if a critical point is a maximum or minimum?

To determine if a critical point is a maximum or minimum, you can use the second derivative test. If the second derivative is positive at the critical point, then it is a minimum. If the second derivative is negative, then it is a maximum.

## 5. Can a function have more than one extremum?

Yes, a function can have multiple extremums. This can occur when the function has multiple critical points, or when the critical point is a saddle point.

Replies
2
Views
552
Replies
7
Views
986
Replies
9
Views
514
Replies
15
Views
541
Replies
2
Views
825
Replies
2
Views
477
Replies
3
Views
343
Replies
9
Views
1K
Replies
10
Views
2K
Replies
9
Views
1K