# Tag Info

### derivative of the max function

No, $g(y)$ is not differentiable in general. For example, let $h: \mathbb{R} \to [0, 1]$ be a smooth function with $h(x) = 0$ when $x \leq 0$ and $h(x) = 1$ when $x \geq 1$. (Such a function exists. ...
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### Interpreting a notation in calculus of variations (differentiating with respect to a derivative)

Consider a functional $J[y]$ defined by: $$J[y] = \int_a^b F(x, y, y') dx \tag{1}$$ I think it will be helpful to remind you right away that your notation implies that you have already agreed to ...
• 556

### Are differentials on their own in stochastic calculus just an abuse of notation?

Differentials in stochastic calculus have very precise interpretations. The stochastic differential equation in differential form $$dX_t = \mu (t, X_t) dt + \sigma (t, X_t) dB_t$$ translates precisely ...
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Accepted

### How to calculate $\int(yy'' + (y')^2)\,dx$?

Even if you couldn't have noticed the $vu' + uv'$, you could still do through integration Integrating through by-parts: \begin{align*} \int yy'' + (y')^2 \, dx &= \int yy'' \, dx + \int (y')^2 \, ...
Accepted

### $|f(x)-f(y)| < |g(x) - g(y)|$ when $|f'| < |g'|$

Fix $x<y$. By Cauchy’s mean-value theorem (which is completely elementary, and is simply a “clever” application of the usual mean-value theorem), there is a number $\xi\in (x,y)$ such that \begin{...
• 55.9k
Accepted

### Finding an increasing function on $\mathbb{R}^+$ such that $f(0)=0$, $\lim_{x\to\infty}f(x)=1$, and $f\circ\left[(f')^{-1}\right]=\frac1{1+x}$

Let $\phi(x)=1/(1+x)$ for $x\in(0,\infty)$ we have $\phi^{-1}(y)=\frac1y-1$ for $y\in(0,1)$. Now $f\circ((f')^{-1})=\phi$ means $(f')^{-1}=f^{-1}\circ\phi$. Take inverse $f'=\phi^{-1}\circ f$. So you ...
• 1,831