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If a function $f(x)$ has a derivative $f'(x)$ then where $f'(x_0) = 0$ there is an extreme point at $x=x_0$. And where $f''(x_0)=0$ there is an inflection point at $x=x_0$.

I am asking are there any significance or applications where $f(x_0)=f'(x_0)$ at some point $x=x_0$?

I am referring to a case like $f(x)=x^2$ and $f'(x)=2x$, they intersect at $(0,0)$ and $(2,4)$ what is special or can be derived from these points?

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  • $\begingroup$ $f(x) = e^x$ is ubiquitous in models of growth or decay $\endgroup$ May 13 '16 at 14:53
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    $\begingroup$ I'm pretty sure this is not a duplicate; the OP is asking what happens if $f(x)=f'(x)$ at some specific point, not on the entire domain. $\endgroup$
    – Micah
    May 13 '16 at 14:56
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    $\begingroup$ It is not correct that there must be an inflection point where $f''(x)=0$. An inflection point is where the graph changes from bending up from the tangent line to bending down from the tangent line (or vice versa). So a sign change in $f''$ is required. For example, if $f(x)=x^4$, then $f''(0)=0$ but $(0,0)$ is not an inflection point (because the graph always lies above the tangent line). $\endgroup$
    – MPW
    May 13 '16 at 15:08
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    $\begingroup$ @MPW Similarly $f'(x_0)=0$ does not imply that $x_0$ is an extremum. $\endgroup$
    – fqq
    May 13 '16 at 15:39
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If I'm interpreting this correctly, I believe you are asking if there is anything special about this happening locally, i.e at a certain point $x_0$, does the fact that $f(x_0)=f'(x_0)$ mean anything? I would say no; note that by just translating the function appropriately, you can make this happen at any point. To see this, for any differentiable $f$, and any point $x_0$ define $g_{x_0}(x)=f(x)+(f'(x_0)-f(x_0))$, and this function will have that property at the point $x_0$.

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Say this happens at the point $(a,b)$. Then the slope of the tangent line that point is also $b$, so the tangent line at that point is $$ y-b=b(x-a) $$ which can be rewritten as $$ y=b(x-(a-1)) $$ That is, the tangent line at $(a,b)$ is the line of slope $b$ which intersects the $x$-axis at the point $(a-1, 0)$.

So, if you linearly approximate $f$ based on its behavior at $(a,b)$, your approximation will vanish one $x$-unit before $a$. In some cases, this might mean your chosen units for $x$ and $f(x)$ are in some way compatible near $(a,b)$. However, it can't mean anything more physical than that, because if you change units then this condition will no longer be true!

A similar question you could ask that might be a little more meaningful (because it does continue to be true after a change of units) is:

What is the significance of a point $x_0$ where $f(x_0)=x_0f'(x_0)$?

By the same kind of computation I did above, you can check that this means the tangent line to $f$ at $x_0$ passes through the origin.

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    $\begingroup$ Relevant to your final comment: for students of economics, this corresponds to AC=MC, i.e. average cost = marginal cost. $\endgroup$
    – πr8
    May 13 '16 at 15:59
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If $f(x_0)=f'(x_0)$, this tells us that if we define $g(x)=f(x)\exp(-x)$, that $g$ has an extreme point at $x_0$ (excepting inflection points etc.). This is because:

$$\frac{dg}{dx}=\frac{d}{dx}f(x)\exp(-x)=f'(x)\exp(-x)-f(x)\exp(-x)$$

$$\implies\frac{dg}{dx}\vert_{x=x_0}=f'(x_0)\exp(-x_0)-f(x_0)\exp(-x_0)=0$$

i.e. $g'(x_0)=0$.

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If $f(x)=f'(x)$, then $f(x) = C \cdot e^x$.
That's the important application/consequence here.

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    $\begingroup$ The OP asks what can we say if $f(x)=f'(x)$ for some $x\in\dom f$, I think, not for every $x$. $\endgroup$ May 13 '16 at 14:55
  • $\begingroup$ @MoebiusCorzer It's not so clear. I understood it for all $x$. $\endgroup$ May 13 '16 at 15:01
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    $\begingroup$ Yes I completely agree with you: it is why I wrote "I think". However, since the OP talks about local properties given by the value of a derivative at a particular point, I assumed it is also what s/he meant in the last part of her/his question. $\endgroup$ May 13 '16 at 15:24
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Well, $f(x) = f'(x)$, for every $x \implies f(x) = Ce^x$, C is a constant that can be determined if you have a inicial condition.

There are many applications, such as radioactive decay or growth of some population!

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