Differentiation question with continuity

In summary, we are given a function f that is continuous and has continuous derivatives of all orders for all x, and it satisfies xf''(x) + f'(x) + xf(x) = 0. We are asked to find the values of f'(0) and f''(0), given that f(0) = 1. From our attempt at a solution, we have found that f'(0) = 0. To find f''(0), we can differentiate the given equation and substitute x=0, which results in 0f''(0) + f'(0) + 0f(0) = 0. Simplifying, we get f''(0) = 0
  • #1
inter060708
24
0

Homework Statement



Suppose a function f is continuous and has continuous derivatives of all orders for all
x. it satisfies xf ''(x) + f '(x) + xf(x) = 0. Given f(0) = 1

find the value of f '(0) and f '' (0).

Homework Equations


The Attempt at a Solution



when x=0,

0f''(0) + f ' (0) + 0f(0) = 0
therefore f ' (0) = 0

Now, I am not sure on how to find f ''(0). Is it undefined?
Thanks for your help.
 
Last edited:
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  • #2
There's no attempt at a solution. You have to make one. Then you can get some help.
 
  • #3
I have added the solution attempt, please help. Thanks :)
 
  • #4
That helps. Try differentiating xf ''(x) + f '(x) + xf(x) = 0 and putting x=0 again.
 
  • #5
Wow. Thanks Dick! :D
 

FAQ: Differentiation question with continuity

What is differentiation?

Differentiation is a mathematical process used to find the rate of change of a function at a specific point. It involves calculating the slope of the curve at that point, also known as the derivative.

What is the relationship between differentiation and continuity?

Differentiation and continuity are closely related concepts in calculus. A function is said to be continuous if it has no breaks or jumps in its graph. When a function is differentiable, it is also continuous. This means that if a function is differentiable at a point, it is also continuous at that point.

What is a differentiability point?

A differentiability point is a point on a function where the derivative exists. In other words, the slope of the function can be calculated at that point. For a function to be differentiable at a point, it must also be continuous at that point.

What is the difference between differentiability and continuity?

The main difference between differentiability and continuity is that differentiability focuses on the rate of change of a function, while continuity focuses on the smoothness of a function. A function can be continuous but not differentiable, but it cannot be differentiable without also being continuous.

What is the chain rule in differentiation?

The chain rule is a rule in calculus used to find the derivative of a composition of functions. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. In other words, it allows us to find the rate of change of a function within a function.

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