Proving Differentiability of f given g'(x) < 0 $\forall$ x

mathwizarddud · 2008-06-28T07:38:50+00:00

Suppose the real valued g is defined on \mathbb{R} and g'(x) < 0 for every real x. Prove there's no differentiable f: R \rightarrow R such that f \circ f = g.

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Start date Jun 28, 2008
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Differentiability

In summary, to prove the differentiability of a function f(x) given g'(x) < 0 for all x, we need to show that the limit of the difference quotient of f(x) exists and that the slope of the tangent line at the point is well-defined. The condition g'(x) < 0 is important as it guarantees the existence of a well-defined tangent line for f(x). However, g'(x) cannot be equal to 0 as this would make it impossible to determine the slope of the tangent line to f(x). Differentiability implies continuity, but the converse is not necessarily true. Other conditions that need to be satisfied include the continuity and differentiability of g(x) at the point in question.

Jun 28, 2008

mathwizarddud

Suppose the real valued [tex]g[/tex] is defined on [tex]\mathbb{R}[/tex] and [tex]g'(x) < 0[/tex] for every real [tex]x[/tex]. Prove there's no differentiable [tex]f: R \rightarrow R[/tex] such that [tex]f \circ f = g[/tex].

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Jun 28, 2008

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1. How do you prove differentiability of a function given g'(x) < 0 for all x?

To prove differentiability of a function f(x) given g'(x) < 0 for all x, we need to show that the limit of the difference quotient of f(x) exists as x approaches a particular value. In other words, we must show that the slope of the tangent line at that point is well-defined.

2. What is the significance of g'(x) < 0 in proving differentiability of f(x)?

The condition g'(x) < 0 means that the function g(x) is strictly decreasing, which implies that the slope of the tangent line to g(x) is negative for all values of x. This condition is important because it guarantees the existence of a well-defined tangent line for f(x) at any point where g(x) is differentiable.

3. Can g'(x) be equal to 0 when proving differentiability of f(x)?

No, g'(x) cannot be equal to 0. This is because if g'(x) = 0, then the function g(x) is not strictly decreasing. In this case, the slope of the tangent line to g(x) is 0, which makes it impossible to determine the slope of the tangent line to f(x) at that point and thus, proving differentiability becomes impossible.

4. What is the relationship between differentiability and continuity in this context?

In this context, differentiability implies continuity. If a function f(x) is differentiable at a point, then it must also be continuous at that point. However, the converse is not necessarily true. A function can be continuous at a point but not differentiable.

5. Are there any other conditions that need to be satisfied to prove differentiability of f(x) given g'(x) < 0 for all x?

Yes, there are other conditions that need to be satisfied. In addition to g'(x) < 0, we also need g(x) to be continuous at the point we are trying to prove differentiability for. Furthermore, we need g(x) to be differentiable on an open interval containing the point. Without these conditions, we cannot guarantee the existence of a well-defined tangent line for f(x) at that point.