Differantiation proof question

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In summary, to prove that there exists a point c on (-1, 1) such that f''(c) >= 1, we use the Fermat's law and the Mean Value Theorem (MVT) twice. We need f to be continuously differentiable on [-1, 1] and twice differentiable on (-1, 1) in order to apply the MVT. This means that f' must be continuous on [-1, 1] and differentiable on (-1, 1). Therefore, we also need f to be continuously differentiable in order to use the MVT for f'.
  • #1
transgalactic
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f(x) is a differentiably continues on [-1,1]
and differentiable twice on (-1,1)
suppose f(0)=-1 and f(1)=1 and function f(x) has on (-1,0) an extreme point.

prove that there is a point c on (-1,1) so f''(c)>=1
??

prove:
because of ferma law
if a function has an extreme point on (-1,0) on the point c1 in (-1,0) then
f(x) differentiable on (c1) and f'(c)=0

because of mean value theorem there is a point c2 on (0,1):
f'(c2)=[f(1)-f(0)]/(1-0)=2
and because f'(x) differentiable once more we can do MVT again
f''(c3)=[f'(c2)-f'(c1)]/c2-c1=(2-0)/(c2-c1)
c1 and c2 are inside (-1,1) so the difference between them is smaller then 2
f''(c3)=2/(c2-c1)>=2/2 >=1

i can't understand why do i need to know that its differentiable continuously
it seem like the fact that it differentiable twice is enough
??
 
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  • #2
To apply the MVT to f' you need that f' is differentiable on (-1, 1) AND continuous on [-1, 1].
This is precisely equivalent to continuously differentiable on [-1, 1] and twice differentiable on (-1, 1), as you can check.
 
  • #3
so we need f(x) to be continuously differentiable in order to do mvt for f'(x)

but we did it twice
we have not been given that f'(x) continuously differentiable too
so we didnt have the right to do mvt on f''(x)
??
 
  • #4
Check again the requirements for the MVT.

Wikipedia said:
Let f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b

We need f to be continuous on the closed interval and differentiable on the open interval. There are no requirements on the continuity of the derivative. So if you want to apply the MVT to f', it is enough if f' is continuous on the closed interval and differentiable on the open interval, which means that f must be continuously differentiable on the closed interval and twice differentiable (without requirement on the continuity of the second derivative) on the open interval.
 
  • #5
thanks :)
 

1. What is differentiation?

Differentiation is a mathematical concept that involves finding the rate of change of a function with respect to its independent variable. It is used to analyze and understand the behavior of functions and is an important tool in many scientific disciplines.

2. How do you prove a differentiation question?

To prove a differentiation question, you need to use the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules help you find the derivative of a function, which is the key step in proving differentiation questions.

3. What are the common mistakes in differentiation proofs?

One common mistake in differentiation proofs is forgetting to apply the chain rule correctly. Another mistake is using the wrong rule for differentiating a function. It is also important to be careful with algebraic manipulations and to avoid making careless errors.

4. How do you know if a differentiation proof is correct?

A differentiation proof is considered correct if the steps followed are mathematically sound and lead to the correct derivative of the given function. It is important to double-check each step and ensure that the final answer matches the expected result.

5. What is the importance of differentiation in science?

Differentiation is crucial in science as it allows us to understand the behavior of complex systems and make predictions about their future states. It is used in fields such as physics, biology, economics, and engineering to model and analyze phenomena and make informed decisions based on the rate of change of variables.

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