# Tagged Questions

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### The second derivative of g(t)= f(x(t), y(t))

For the chain rule for differentiating a function $g(t) = (f(x), y(t))$ how do you get from the identity $g'(t) = \frac{df}{dx} \cdot \frac{dx}{dt}+ \frac{df}{dy}\cdot\frac{dy}{dt}$ to the identitiy ...
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### A function with positive Hessian at a critical point, without having a minimum there

I have a problem with a little instance: $f(x,y) = \begin{cases} (x^4-3x^2y^2+y^2)/(x^2+y^2) & otherwise \\ 0 & \text{(x,y)=(0,0)} \end{cases}$ This is a example of a function which ...
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### Quasi-concavity of a function of two variables such as $z=(x^a + y^b)^2$

If I have a function such as $z=(x^a + y^b)^2$ with $a$ and $b$ both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative? The ...
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### Finding the critical points of a function

What are the steps in finding the critical points of a function in general? Say for example, the function $$f(x, y) = 2x^3 + 11x^2 + 0.5y^2 - 2xy$$ I can't quite seem to understand the steps/method ...
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### find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
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### Continuosly differentation on composite functions

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ a $C^1$ function and defined $g(x) = f(\|x\|)$. Prove $g$ is $C^1$ on $\mathbb{R}^n\setminus\{0\}$. Give an example of $f$ such that $g$ is $C^1$ at the origin ...
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### degree of homogeneity

I have the function $$f(x,y)=\frac{y^b}{x^a}+\frac{x^b}{y^a}\quad a,b\gt0$$ The questions I have to answer are For which a and b is the function homogenous? Determine the degree of homogeneity My ...
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### Question regarding Continuity of F(x,y)

Let $f(x,y) = \begin{cases} \frac{2(x^3+y^3)}{x^2+2y}&\text{ } (x,y)\not=(0,0)\\ 0 &\text{ }(x,y) =(0,0). \end{cases}$ show that first order partial derivatives of $f$ wrt x and y exist at ...
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### What is the domain and range of this multivariable function?

What is the domain and range of the following multivariable function? $g(x,y,z) = {1\over \sqrt{(4 - x^2 - y^2 - z^2)}}$ g is real valued. So far I have that the domain is the set of a vectors ...
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### Variants of the bump function.

The title of this question isn't really clear because of the 150 char limit. What I actually want to ask is this: If I would have a bump function for $-1 < x < 1$ and I would have some ...
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### Prove $f(x,y) = xy/(x^2 + y^2)$ is continuous everywhere except $(0,0).$

I'd just like to ask you if my proof here is valid. I'll provide you with the method I used and if it seems ok let me know! If not, explanations would be helpful! My main approach to this question ...
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### What is the difference between these two functions?

$$r(x) = \langle x, x^2-1 \rangle$$ $$f(x)=x^2-1$$ Their graph is the same, but one is called vector valued function while the other one is a regular one. I think I'll never get to understand this ...
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### Boundedness of an implicit function

I have a function $h(u) : \mathbb{R}^N \rightarrow \mathbb{R}^N$ in an implicit form: $$A(h, u) = 0.$$ What conditions could we put on $A$ to guarantee that $h$ is bounded? And if we have an ...
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### differentiability and linear operators

I could give suggestions for this question. $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is differentiable in $a$, then for each $h \in \mathbb{R}^{n}$ exists a linear transformation ...
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### limit images Ball $B[a,r]$

Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ continuous function. I have difficulty proving formally that meets $$\lim\limits_{r \to 0 } f(B[a,r]) =f(a)$$ where $B[a,r]$ is the closed ball in ...
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### looking for a diffeomorphism (not C1)

Let $f\colon\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ diffeomorphism with $f(B[0,1])\subset B[0,1]$ and $| \det f^{\prime}(x) |<1/2$ for all $x\in B[0,1]$ then for every continuous function ...
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### J-measurable sets and functions of class $C^1$

If $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is $C^1$ class and $det f^{\prime}(0)=0$ show that, when $r\rightarrow 0$ $$\dfrac{Vol(f(B[0,r]))}{ Vol(B[0,r])} \rightarrow 0$$ where $Vol(X)$ is ...
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### Quesiton about functions

I'm embarrassed asking this question. I took a long break. main question is : cartesian coordinate system in $R^n$ space is shown as $(x_1,x_2 ...x_n)$. Show that for $1\le i\le n$ each ...
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### Determining if the product of two particular harmonic functions is a harmonic function

Let $u$ be a $C^{2}$ harmonic function in $\mathbb{R}^{n}$ and let $g(x) = \left| x \right|^{2-n}$. I would like to show that: $v(x) = g(x)u\left(\frac{x}{\left| x \right|^{2}}\right)$ is also ...
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### Discontinuity in multivariable calculus

Show that the function F(x, y) = x(1 - cos(x - y))/(x y)^2 has removable discontinuity along the line x = y. Which values should we assign to this function on the diagonal x = y in order to turn it ...
### Prove function (0, ∞) to (0, ∞) can't exist if $(f(x))^2>(f(x+y))((f(x)+y))$
So we're trying to prove that no function exists $f: (0,\infty) \rightarrow(0,\infty)$ such that $f(x)^2\ge f(x+y) (f(x)+y)\quad x,y\gt0$ I've tried to break it up into 3 cases to show if $x\gt y$ ...