Differentiability and functional equations

In summary, the homework statement is telling you that there is a function that takes in two real numbers and returns a real number, and that the first order derivative of the function is -1. You are also told that the function exists and is equal to -1. The next step is to find the function's derivative, which you do by using the basic formula of differentiablilty. The next step is to find the function's limit, which you do by using the given functional equation and substituting in the values for x and y. Finally, you are asked to find the function's value at x=2, but you are not given the information needed to do so.
  • #1
Tanishq Nandan
122
5

Homework Statement


Let f((x+y)/2)= {[f(x)+f(y)]/2} for all real x and y
{f'(x)=first order derivative of f(x)}
f'(0) exists and is equal to -1 and f(0)=1.
Find f(2)

Homework Equations


Basic formula for differentiablilty:
f'(x)=limit (h tends to 0+) {[f(x+h)-f(x)]/h}

The Attempt at a Solution


I know that when you have a functional equation along with some info about it's derivative,you need to apply the basic formula of differentiablilty to find f'(x) and evaluate the limits using the given functional equation..and that's precisely what I did..but how do I get to f(2) from this??
20170701_230508-1.jpg
Hope you get what I did in the second last step..I used the given functional equations,placed x=2h and y=0,then replaced the value of f(h) obtained from there in the given limit.
Now,as I said,how to get f(2) from here??
 
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  • #2
Thread has been moved. Please post questions that involve derivatives in the Calculus & Beyond section.
Tanishq Nandan said:
Let f((x+y)/2)= {[f(x)+f(y)]/2} for all real x and y
{f'(x)=first order derivative of f(x)}
f'(0) exists and is equal to -1 and f(0)=1.
Find f(2)
I don't think your approach using the definition of f'(0) will help you. From the given information, ##\frac 1 2 \left(f(1) + f(-1)\right) = 1##, or equivalently
##f(1) + f(-1) = 2##. In addition, ##f(1/2) + f(-1/2) = 2, f(2) + f(-2) = 2##, and so on. Numbers that are opposites always have function values that add up to 2. What sort of curves have this property?
 
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  • #3
The function is odd about x=1 ! !
f(2) will correspond to negative of f(0)
Ans (-1) (and the answer is matching)
But,just one slight problem..where did we make use of f'(0)=-1 ? ?
 
  • #4
Tanishq Nandan said:
The function is odd about x=1 ! !
f(2) will correspond to negative of f(0)
Ans (-1) (and the answer is matching)
But,just one slight problem..where did we make use of f'(0)=-1 ? ?

[tex]f\left(\frac{(1 + x) + (1 - x)}{2}\right) = \frac{f(1 + x) + f(1-x)}{2}[/tex] gives you [itex]f(1 + x) + f(1-x) = 2f(1)[/itex], which doesn't help you as you don't know what [itex]f(1)[/itex] is.

Your functional equation is actually telling you that [itex]f[/itex] is both concave and convex. What functions have that property?
 
  • #5
A straight line?
Y=mx+c??
 

FAQ: Differentiability and functional equations

1. What is the definition of differentiability?

Differentiability is a mathematical property that describes the smoothness or continuity of a function. A function is differentiable at a point if the limit of the slope (or rate of change) of the function at that point exists. This means that the function has a well-defined tangent line at that point.

2. How is differentiability related to continuity?

Differentiability and continuity are closely related concepts. A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. If a function is differentiable at a point, it must also be continuous at that point. However, a function can be continuous at a point without being differentiable at that point.

3. What is the difference between a differentiable function and a continuous function?

A differentiable function is a function that has a well-defined derivative at every point in its domain. This means that the function is smooth and has a well-defined slope at every point. A continuous function, on the other hand, only needs to have a well-defined limit at every point in its domain.

4. What are some common examples of functional equations?

Some common examples of functional equations include the Cauchy-Riemann equations, the Euler-Lagrange equations, and the Navier-Stokes equations. These equations are used to describe different types of relationships between variables and are often used in physics, engineering, and other scientific fields.

5. How are functional equations used in real-world applications?

Functional equations are used in many real-world applications, such as modeling physical phenomena, optimizing systems, and predicting outcomes. For example, the Navier-Stokes equations are used to model fluid flow in pipes and can be used to design more efficient systems. The Euler-Lagrange equations are used in optimization problems to find the most efficient solution. Functional equations are also used in economics, biology, and other fields to model relationships between variables and make predictions.

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