Proving Differentiability Using Inequalities

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In summary, using the squeeze theorem and the given conditions, we can prove that g(x) is differentiable at a and that g'(a) = g'(a) = h'(a). This is because the derivatives of f(x) and h(x) at a are equal, and the inequalities involving g(x) can be rewritten to show that g'(a) falls between f'(a) and h'(a). Therefore, g(x) satisfies the conditions for being differentiable at a and its derivative at a is equal to f'(a) and h'(a).
  • #1
redyelloworange
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Suppose
f(a) = g(a) = h(a)
and f(x) <= g(x) <= (x) for all x

Prove g(x) is differentible and that
f'(a) = g'(a) = h'(a).

So.. I need to prove that the following limit exists:

lim h -->0 (g(x+h) - g(x)) / h

but how can i use the fact that f(x) <= g(x) <= (x) for all x?

Thanks
 
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  • #2
Obviously you're going to have to tell us something about f and h.
 
  • #3
f(a) = g(a) = h(a)
and f(x) <= g(x) <= h(x) for all x

and

f'(a) = h'(a)
 
Last edited:
  • #4
Is it possible that f and h are given as differentiable? Otherwise, just take f(x)= g(x)= h(x) to be any non-differentiable functions and the hypotheses are satisfied while the conclusion is not true!
 
  • #5
you're right

*I seem to have typed the question wrong many times*
sorry.

So, given:

f(a) = g(a) = h(a)
and
f(x) <= g(x) <= h(x) for all x
and
f'(a) = h'(a)

Prove g is differentiable at a, and that g'(a) = g'(a) = h'(a).
 
  • #6
Do you know the squeeze theorem?
 
  • #7
oh!
sol instead of just including g(x) in the inequality.. i include more:

f(x) <= g(x) <= h(x)
f(a+h) <= g(a+h) <= h(a+h)
f(a+h) - f(a) <= g(a+h) - f(a) <= h(a+h) - f(a)
f(a+h) - f(a) <= g(a+h) - f(a) <= h(a+h) - f(a)

f'(a) <= g'(a) <= h'(a)

but f'(a) = h'(a)

so (how exactly do i phrase this?)

f'(a) = g'(a) = h'(a)

ooh. that was a good hint/spark =) thanks!
 

Related to Proving Differentiability Using Inequalities

1. What is the Spivak Derivative Question?

The Spivak Derivative Question, also known as the "calculus conundrum", is a mathematical problem posed by mathematician Michael Spivak in his book "Calculus". It asks for a function that is continuous everywhere but differentiable nowhere.

2. Why is the Spivak Derivative Question important?

The Spivak Derivative Question is important because it challenges our understanding of calculus and the fundamental concept of differentiability. It also has practical applications in real-world problems, such as finding functions that model real-life situations where smoothness is desired but not differentiability.

3. Has anyone solved the Spivak Derivative Question?

No, the Spivak Derivative Question remains unsolved to this day. Many mathematicians have attempted to find a solution, but none have been able to prove the existence of a function that is continuous but not differentiable everywhere.

4. What are some potential solutions to the Spivak Derivative Question?

Some potential solutions proposed by mathematicians include fractal functions, which have continuous but non-smooth graphs, and functions that are differentiable at every point except one, known as the Weierstrass function. However, these solutions have not been proven to be the answer to the Spivak Derivative Question.

5. How does the Spivak Derivative Question relate to other unsolved mathematical problems?

The Spivak Derivative Question is often compared to other unsolved problems in mathematics, such as the Riemann Hypothesis and the Collatz Conjecture. These problems challenge our current understanding of mathematics and have sparked numerous attempts at finding solutions. However, much like the Spivak Derivative Question, these problems remain unsolved and continue to fascinate and intrigue mathematicians around the world.

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