Local Extremum of f(x)=(4-x^2)^(-1/2): First Derivative Test

In summary, a local extremum is a point on a graph where a function reaches either a maximum or minimum value within a small neighborhood. To find a local extremum using the first derivative test, one must find the first derivative of the function, set it equal to 0, and determine the sign of the derivative on either side of the critical points. The second derivative test can then be used to confirm the nature of the extremum. A function can have multiple local extrema, and there are other methods besides the first derivative test to find them, such as graphing, optimization, or using the second derivative test.
  • #1
MIA6
233
0
1. According to the First Derivative Test for local extrema, if f' doesn't change sign at c, then f has no local extreme value at c. But for a question on my book, f(x)=(4-x^2)^(-1/2), the critical point is 0, but i think it doesn't have local extreme because the derivative doesn't change sign ,but my book says it has minimum value at x=0.
Thanks for help.
 
Physics news on Phys.org
  • #2
Can you write out the calculations leading you to conclude that the derivative doesn't change sign at x = 0?
 
  • #3


The First Derivative Test is a useful tool for determining local extrema, but it is not always a definitive answer. In the case of f(x)=(4-x^2)^(-1/2), the critical point at x=0 does not change sign of the derivative, but this does not necessarily mean that there is no local extreme at that point. The First Derivative Test is only a necessary condition for a local extreme, not a sufficient one. In this case, we need to look at the second derivative to determine the nature of the critical point at x=0. If the second derivative is positive, then we can conclude that there is a local minimum at x=0. However, if the second derivative is negative, then there is a local maximum at x=0. It is important to remember that the First Derivative Test is not a guarantee for local extrema, and further analysis may be needed in some cases.
 

1. What is a local extremum?

A local extremum is a point on a graph where the function reaches either a maximum or minimum value within a small neighborhood of that point. It is also known as a local maximum or local minimum.

2. How do you find the local extremum of f(x)=(4-x^2)^(-1/2) using the first derivative test?

To find the local extremum using the first derivative test, you need to follow these steps:

  • Find the first derivative of the function, which is f'(x) = 2x/(4-x^2)^(3/2).
  • Set f'(x) equal to 0 and solve for x. This will give you the critical points where the function may have a local extremum.
  • Plot these critical points on a number line and determine the sign of f'(x) on either side of each point.
  • If the sign changes from positive to negative or vice versa at a critical point, then that point is a local extremum.
  • Use the second derivative test to confirm whether the point is a local maximum or local minimum.

3. What is the second derivative test and how does it confirm the nature of the local extremum?

The second derivative test involves finding the second derivative of the function and evaluating it at the critical point. If the second derivative is positive, then the point is a local minimum. If the second derivative is negative, then the point is a local maximum. If the second derivative is 0, then the test is inconclusive and other methods must be used to determine the nature of the extremum.

4. Can a function have more than one local extremum?

Yes, a function can have multiple local extrema, including both local maxima and minima. These points may occur at different locations on the graph.

5. Is the first derivative test the only way to find local extrema?

No, there are other methods such as the second derivative test, graphing the function, or using calculus techniques such as optimization to find local extrema. The first derivative test is just one way to determine the presence and nature of local extrema.

Similar threads

MHB Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x
    • Calculus
Replies
1
Views
1K
I Understanding a proof of inexistence of max nor min
    • Calculus
Replies
4
Views
883
I Derivation of a Higher Order Derivative Test
    • Calculus
Replies
5
Views
2K
I Partial derivatives of the function f(x,y)
    • Calculus
Replies
6
Views
1K
I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
    • Calculus
Replies
9
Views
851
MHB -x4.6 Find extrema f(x, y, z) = x + yz
    • Calculus
Replies
4
Views
2K
MHB 243.14.7.4 Find all local extreme values
    • Calculus
Replies
2
Views
1K
MHB Where are the one-sided derivatives of f(x) equal?
    • Calculus
Replies
2
Views
711
B Second derivatives and inflection points
    • Calculus
Replies
5
Views
2K
B How do I invert this exponential function?
    • Calculus
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top