Solving a Complex Analysis Problem: Finding Critical Points of k(x)

In summary, the conversation discusses finding critical points of a function that involves a third degree polynomial and a fraction. The first derivative test is used to classify these critical points. The speaker also mentions that this topic may be too complex for an analysis assignment.
  • #1
Treadstone 71
275
0
"Let a,b be in R with a>0 and f(x)=ax^3+bx. Let k(x)=[f''(x)]/[1+(f'(x))^2]^(3/2). Find the critical points of k(x) and use the first derivative test to classify them."

This seems incredibly quantitative and complicated for an analysis assignment. There must be a theorem of some kind I can use to solve this, but I can't see it.
 
Physics news on Phys.org
  • #2
I used to think that expressions like "such and such is too complicated for an analysis assignment" just shouldn't be used.

Now that I'm in my second year of analysis courses, I would like my professor to know that expressions like "such and such is too complicated for an analysis assignment" should be employed far more often :smile: .
 
Last edited:

FAQ: Solving a Complex Analysis Problem: Finding Critical Points of k(x)

What is a critical point in complex analysis?

A critical point in complex analysis is a point where the derivative of a complex-valued function is equal to zero. In other words, it is a point where the function has a horizontal tangent line and can potentially be a maximum, minimum, or saddle point.

How do you find critical points in complex analysis?

To find critical points in complex analysis, you can use the Cauchy-Riemann equations to express the derivative of a complex-valued function in terms of its partial derivatives. Then, set both the real and imaginary parts of the derivative equal to zero and solve for the complex variable.

What is the significance of critical points in complex analysis?

Critical points play a crucial role in determining the behavior of a complex-valued function. They can help identify local extrema and saddle points, which in turn can provide information about the overall behavior of the function.

Can all complex-valued functions have critical points?

No, not all complex-valued functions have critical points. Functions that are constant or have an infinite derivative at every point do not have critical points. Additionally, some functions may have critical points only in certain regions of their domain.

How can critical points be used to solve complex analysis problems?

Critical points can be used to find local extrema and saddle points of complex-valued functions, which can aid in solving optimization problems and understanding the behavior of the function. They can also be used to determine the convergence or divergence of power series and contour integrals.

Similar threads

Why can't a critical point of a system of DEs be complex?
Replies
27
Views
1K
Odd/even function and critical points
Replies
4
Views
2K
Solve integral by finding Fourier series of complex function
Replies
3
Views
921
Solving Multivariate Problem: Critical Points for ##y=-x##
Replies
2
Views
1K
[Complex analysis] Contradiction in the definition of a branch
Replies
8
Views
2K
Newton's method to approximate critical point
Replies
9
Views
3K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
Replies
3
Views
1K
Lagrange multipliers understanding
Replies
8
Views
1K
Prove that this series converges uniformly on R
Replies
4
Views
1K
Solving System of Equations - Critical Points
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top