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

Click For Summary
SUMMARY

The discussion centers on the function f(x)=(4-x^2)^(-1/2) and its critical point at x=0. According to the First Derivative Test, if the derivative f' does not change sign at a critical point, then there is no local extremum at that point. However, the book asserts that there is a minimum value at x=0. Participants in the discussion seek clarification on the calculations that demonstrate the behavior of the derivative at this critical point.

PREREQUISITES
  • Understanding of calculus concepts, specifically the First Derivative Test.
  • Familiarity with critical points and local extrema.
  • Knowledge of how to compute derivatives of functions.
  • Ability to analyze the behavior of functions around critical points.
NEXT STEPS
  • Review the First Derivative Test in detail.
  • Practice calculating derivatives for various functions, including f(x)=(4-x^2)^(-1/2).
  • Explore the concept of local extrema and how they are determined.
  • Investigate the implications of non-changing sign derivatives at critical points.
USEFUL FOR

Students studying calculus, educators teaching derivative concepts, and anyone interested in understanding local extrema and their determination through the First Derivative Test.

MIA6
Messages
231
Reaction score
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
Can you write out the calculations leading you to conclude that the derivative doesn't change sign at x = 0?
 

Similar threads

MHB Is f(x)=ln(x)/x Increasing or Decreasing?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Derivation of a Higher Order Derivative Test
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad ##f(2x)=f^2(x)-2f(x)-1/2## then find ##f(3)##
  • · Replies 17 ·
Replies
17
Views
4K
Undergrad Understanding a proof of inexistence of max nor min
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Partial derivatives of the function f(x,y)
  • · Replies 6 ·
Replies
6
Views
3K
MHB -x4.6 Find extrema f(x, y, z) = x + yz
  • · Replies 4 ·
Replies
4
Views
2K
MHB 243.14.7.4 Find all local extreme values
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Proving that convexity implies second order derivative being positive
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
1K
Undergrad Second Derivative Test for Local Extrema
  • · Replies 3 ·
Replies
3
Views
2K