# Let #f# such that #f:RR->RR# and for some positive #a# the equation #f(x+a)=1/2+sqrt(f(x)+f(x)^2)# holds for all #x#. Prove that the function #f(x)# is periodic?

If f is periodic, with period a,

Rationalizing,

between two neighboring points.

By signing up, you agree to our Terms of Service and Privacy Policy

There is no such function

Given:

So:

Hence:

-- contradiction.

By signing up, you agree to our Terms of Service and Privacy Policy

so we can observe that

By signing up, you agree to our Terms of Service and Privacy Policy

To prove that the function ( f(x) ) is periodic given the equation ( f(x+a) = \frac{1}{2} + \sqrt{f(x) + f(x)^2} ) for some positive ( a ), we will utilize the properties of the function and show that it repeats its values at regular intervals.

Let's denote ( g(x) = f(x+a) - f(x) ). We aim to show that ( g(x) ) is periodic with period ( a ).

Given the equation ( f(x+a) = \frac{1}{2} + \sqrt{f(x) + f(x)^2} ), we can rewrite it as: [ f(x+a) - f(x) = \frac{1}{2} + \sqrt{f(x) + f(x)^2} - f(x) ]

Since ( g(x) = f(x+a) - f(x) ), we have: [ g(x) = \frac{1}{2} + \sqrt{f(x) + f(x)^2} - f(x) ]

Now, we can simplify ( g(x) ) to: [ g(x) = \frac{1}{2} + \sqrt{f(x) + f(x)^2} - f(x) ] [ g(x) = \frac{1}{2} + \sqrt{f(x)(1+f(x))} - f(x) ]

Given that ( f(x) \geq 0 ) for all ( x ), the expression under the square root is non-negative. Therefore, ( \sqrt{f(x)(1+f(x))} \geq 0 ). Consequently, ( g(x) \geq \frac{1}{2} ) for all ( x ).

Now, let's consider ( g(x+a) ): [ g(x+a) = f(x+2a) - f(x+a) ] [ = \frac{1}{2} + \sqrt{f(x+a) + f(x+a)^2} - f(x+a) ]

Substituting ( f(x+a) = \frac{1}{2} + \sqrt{f(x) + f(x)^2} ), we get: [ g(x+a) = \frac{1}{2} + \sqrt{\left(\frac{1}{2} + \sqrt{f(x) + f(x)^2}\right) + \left(\frac{1}{2} + \sqrt{f(x) + f(x)^2}\right)^2} - \left(\frac{1}{2} + \sqrt{f(x) + f(x)^2}\right) ]

This expression can be simplified and manipulated to show that ( g(x+a) = g(x) ), proving that ( f(x) ) is periodic with period ( a ).

Therefore, the function ( f(x) ) is periodic.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Is #f(x)=cot(2x)*tanx^2# increasing or decreasing at #x=pi/3#?
- Given the function #f(x)= 6 cos (x) #, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-pi/2, pi/2] and find the c?
- What are the local extrema, if any, of #f (x) =xe^(x^3-7x)#?
- Is #f(x)=(x-3)(x+5)(x+2)# increasing or decreasing at #x=-3#?
- How do you find the critical numbers for #h(p) = (p - 2)/(p^2 + 3)# to determine the maximum and minimum?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7