Prove Max/Min of f(x) -> Sqrt of f(x) at x0

  • Thread starter Thread starter minia2353
  • Start date Start date
  • Tags Tags
    Maximum Minimum
AI Thread Summary
If f(x) is non-negative on an interval and has a maximum at x0, then the square root of f(x) also achieves a maximum at x0. This is because if f(x0) is the maximum, then for any y in the interval, f(y) is less than or equal to f(x0). Since the square root function is increasing for non-negative values, it follows that sqrt(f(y)) is less than or equal to sqrt(f(x0)). The same reasoning applies to minimum values, confirming that sqrt(f(x)) will also have a minimum at x0. Thus, the proof hinges on the properties of the square root function and the behavior of f(x) at its extremum.
minia2353
Messages
13
Reaction score
0

Homework Statement



prove: if f(x) bigger or the same as 0 on an interval I and if f(x) has a maximum value on I at x0(0 is written small beside the x), then sqrt of f(x) also has a maxsimum value at x0. Similarily for minimum values. Hint: Use the fact that sqrt of x is an increasing function on the interval zero to plus infinity.

Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
Looks pretty direct to me. If f(x_0) is a maximum for f, then f(y)\le f(x_0) for all y.Since square root is an increasing function, \sqrt{f(y)}\le \sqrt{f(x_0)}.
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
View full post »

Thread 'Making the denominator of a certain fraction real'

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
View full post »

Similar threads

Find the max and min value from equation?
Replies
13
Views
2K
Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
Replies
3
Views
578
Python Finding a local max/min of a function in Python
Replies
7
Views
5K
Equation for the amplitude of a line sweeping across a triangle
Replies
6
Views
3K
Composite and Inverse Functions Problem
Replies
3
Views
2K
I Understanding a proof of inexistence of max nor min
Replies
4
Views
2K
Proving period of f/g could be less than period of f and g
Replies
20
Views
7K
Minimum of x+1/x (Perimeter Square < Perim equal area rects)
Replies
1
Views
2K
Finding the Max/Min of 1/f(x): A Proof
Replies
11
Views
4K
F(x, y) Min Max problem with boundaries
Replies
30
Views
3K

Hot Threads

Recent Insights

Back
Top