New answers tagged

0 votes

Show that $f(x)=\cos x-\sin x$ is bounded

Never subtract two inequalities which are in the same direction It some time you get right result and wrong other time. So subtraction of inequalities is a spurious exercise as the subtraction is non-...
user avatar
  • 39.3k
3 votes

Show that $f(x)=\cos x-\sin x$ is bounded

Just apply the triangle inequality: $$\lvert \cos x-\sin x\rvert\leq\lvert \cos x\rvert +\lvert \sin x\rvert\leq 2.$$
user avatar
  • 3,521
1 vote

Show that $f(x)=\cos x-\sin x$ is bounded

You can write $$\begin{cases}-1\le\cos x\le1\\{-1}\le\sin x\le1\end{cases}\\ \implies \begin{cases}-1\le\cos x\le1\\{1}\ge-\sin x\ge -1\end{cases} $$ We see that it is not possible to sum these ...
user avatar
  • 973
2 votes

Show that $f(x)=\cos x-\sin x$ is bounded

In this way you can find "exact bound" (least upper bound) $$\cos(x)-\sin(x)=a$$ $$\frac{\cos(x)}{\sqrt{2}}-\frac{\sin(x)}{\sqrt{2}}=\frac{a}{\sqrt{2}}$$ $$\cos\left(\frac{\pi}{4}\right)\cos(...
user avatar
1 vote

Finding the minimum value of the expression $xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2$

An approach with symmetric functions: $$A=xy(1-z)^2+xz(1-y)^2+yz(1-x)^2\\ \quad =xy+xz+yz-6xyz+xyz(x+y+z)\\ \quad =xy+xz+yz-5xyz$$ the symmetric $\quad xyz\quad$ in terms of $\quad x^3+y^3+z^3:$ $(x+y+...
user avatar
1 vote

Let $f \in C^{\infty}(\mathbb{R}^n)$ be such that $\underline{0}$ is a local minimum point for every algebraic curve, is it $0$ a local minimum point?

To get you started: Let $g$ be smooth and nonnegative with $g^{(k)}(0)=0$ for all $k=0,1,2,\dots$. For example, take $$g(t) = \begin{cases} e^{-1/t^2}, & t\ne 0 \\ 0, & t=0\end{cases}.$$ Write ...
user avatar
2 votes
Accepted

Let $f \in C^{\infty}(\mathbb{R}^n)$ be such that $\underline{0}$ is a local minimum point for every algebraic curve, is it $0$ a local minimum point?

Probably we can give an explicit example, but being lazy we can use Whitney extension theorem to only have to talk about some of the values at some points or sets of points, and the rest gets filled ...
user avatar
1 vote

Finding the minimum value of the expression $xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2$

You have not used the condition of $x+y+z=1$. Since when $x=y=z=1/3$ the equality holds, your answer is the same with the $4xyz$ answer. So the textbook answer is flawed. you can get a numerical ...
user avatar
  • 2,698
0 votes

Choosing a cost function for optimization

I have an idea. Let's say that all three requirements apply. Then $-\log(1-x)$ is an increasing function of $x$, and both it and its derivative increase without bound as $x\rightarrow 1^-$. So it's ...
user avatar
0 votes

Finding global maxima and global minima of a discrete integer valued function

As $x, y\le 127$ and both $z$ and $255-z$ are non-negative, we have: $\begin{align}f(x,y,z)&=xz+y(255-z)&\\&\le 127z+127(255-z)\\&=127\cdot 255\\&=32385\end{align}$ This maximum is ...
user avatar
2 votes
Accepted

Finding the maximum distance from origin of a point on the curve $x=a\sin t-b\sin\left(\frac{at}b\right), y=a\cos t-b\cos\left(\frac{at}b\right)$

Let $D(t)$ be the distance of point P from origi On. Then $$OP=D(t)=\sqrt{a^2+b^2-2ab\cos c t}, c=(b-a)/a,~ a,b>0$$ $D(t)$ will admit maximum value when $ct=\pi$, hence $D_{max}=a+b.$
user avatar
  • 39.3k
1 vote

Smallest value of $b$ when $0<\left\lvert \frac{a}{b}-\frac{3}{5}\right\rvert\leq\frac{1}{150}$

Once you get that $\frac{a}{b}$ is between $\frac{89}{150}$ and $\frac{91}{150}$, you may simply consider the continued fractions of these rational numbers: $$ \frac{89}{150}=[0;1,1,2,5,1,1,2] $$ $$ \...
user avatar
0 votes

Smallest value of $b$ when $0<\left\lvert \frac{a}{b}-\frac{3}{5}\right\rvert\leq\frac{1}{150}$

HINT: If $\frac{a}{b}$ satisfies the condition then $$|5 a - 3 b| = 1$$ the explanation being that in the parallelogram on the vectors $(3,5)$ and $(a,b)$ there should be no other integer points ...
user avatar
  • 47.8k
2 votes

Finding the minimum value of the expression $xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2$

Note that, If $x,y\to 0$ with $z\to 1$, we have $$\inf \left\{xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\mid x+y+z=1 \wedge x>0\wedge y>0\wedge z>0\right\}=0$$ As for obtaining the inequality $A≥4xyz$,...
user avatar
  • 973
2 votes
Accepted

In general what are the min and max values of $(x)_{\pmod d} - \dfrac{(x)_{\pmod {2d}}}{2}$?

For $0\le x<d$, $f(x)=x-\dfrac{x}{2}=\dfrac{x}{2}$; for $d\le x<2d$, $f(x)=(x-d)-\dfrac{x}{2}=\dfrac{x}{2}-d$. Also notice that $2d$ is a period of $f(x)$, so the min and max values are $-\dfrac{...
user avatar
  • 68
6 votes
Accepted

Minimum value of $\sum_{n=0}^\infty\frac{\cos nx}{3^n}$?

The series can be exactly evaluated by passing to the real part of the geometric series in $e^{inx}/3^n$. You get: $$\sum_{n\ge0}\frac{\cos nx}{3^n}=\frac{1}{2}+\frac{2}{5-3\cos x}$$From which the ...
user avatar
  • 11.9k
2 votes

If $|ax^2+bx+c|\leq 2\ \ \forall x\in[-1,1]$ then find the maximum value of $|cx^2+2bx+4a|\ \ \forall x\in [-2,2]$.

We have \begin{align*} |a - b + c| &\le 2, \tag{1}\\ |a + b + c| &\le 2, \tag{2}\\ |c| &\le 2. \tag{3} \end{align*} Using (1) and (2), we have $$|a + c| + |b| \le 2. \tag{4}$$ (Note: If ...
user avatar
  • 24.8k
0 votes

Solve a problem to minimise a cost using derivatives

Some observations: The radius of the tank, $r$, must be $\leq 7$ (can you tell why?). In order to comply with the height restriction, we must have that $ r \ge \sqrt{\dfrac{1000}{11\pi}}\approx 5.38 $...
user avatar
  • 16.4k
2 votes

Finding $c$ such that $\ln\cosh x \approx \sqrt{c^2+x^2}-c$

This is not an answer. Numerical results Consider the norm $$\Phi(c,k)=\int_0^{10^k} \Big[\sqrt{c^2 +x^2}-c -\log (\cosh (x))\Big]^2\,dx$$ and minimize it with respect to $c$ for a given $k$ $$\left( \...
user avatar
2 votes
Accepted

A new infinite series for the minimum of the Gamma function?

Assume arbitrary constants $x_0,\xi\in\textbf{R}$. Then exists constants $a_0,a_1,a_2,\ldots$ and analytic function $f(x)$ such that (take for example $a_k=\left(\frac{x_0-1}{\xi x_0}\right)^k$): $$ ...
user avatar
0 votes

If $|ax^2+bx+c|\leq 2\ \ \forall x\in[-1,1]$ then find the maximum value of $|cx^2+2bx+4a|\ \ \forall x\in [-2,2]$.

We may also examine the behavior of the polynomials in absolute-value brackets as described in "vertex form", $$ p(x) \ \ = \ \ a·\left(x \ + \ \frac{b}{2a} \right)^2 \ + \ \left(c \ - \ \...
user avatar
  • 2,531
1 vote

Maximum-Minimum value of sum of modulus of complex numbers

The given inequality bounds in the question were wrong The correct bounds will be $\sqrt{3}$ and $13\over{4}$ Let me continue my solution Let t = $ |1+z|$ $t^2 = |1+z|^2 = (1+z)\cdot(1+\bar{z})$ = $1+\...
user avatar
4 votes

Evaluate $\lim\limits_{n\to\infty}\big(\frac{n}{2}+\min\limits_{x\in\mathbb{R}}\sum_{k=0}^n{\cos(2^k x)}\big)$

Long comment. I believe this type of question is extremely hard, if not impossible by the current technology, to answer. However, let me share some observation. Define $\varphi_n$ by the $n$th sum: $$ ...
user avatar
0 votes

Evaluate $\lim\limits_{n\to\infty}\big(\frac{n}{2}+\min\limits_{x\in\mathbb{R}}\sum_{k=0}^n{\cos(2^k x)}\big)$

This is far from an answer. The goal is to simplify the expression and to connect this problem with ergodic theory. We first note that w.l.o.g, we can assume $x\in[0,\pi]$. Denote $y=x/\pi$. Let the ...
user avatar
0 votes

Conjecture about the minimum of the Gamma's function

I give here some further terms $a_k$ we have graphically : $$f\left(x\right)=\frac{\left(x!^{1+x!^{\left(\frac{1}{2}+x!\right)^{\left(\frac{1}{4}+x!\right)}}}+x!^{1+x!^{\left(\frac{1}{2}+x!\right)^{\...
user avatar
  • 3,160
0 votes

find maximum of product of two numbers.

Simplifying everything we can, $$l \times t =\left(\frac{100}{x + 3}\right)\left\lfloor\frac{x + 2}{4}\right\rfloor$$ and this is similar to a rational function. To maximize this value as much as we ...
user avatar
  • 1,974
2 votes
Accepted

Find the maximum value of the integral $\int_{-1}^1|x-a|e^xdx$ where $|a|\le1$

Start from your step: $f'(a)=e^a-e^{-1}-(e-e^a)$ $$f''(a)=2e^a>0$$ So your $a=\ln(\frac{e^2+1}{2e})$ is local minima, not maxima, the function is concave up with respect to $a$, so you will find ...
user avatar
  • 5,400
0 votes

How to find global maxima in open intervals

The function is increasing in $(0,1)$ and decreasing in $(1,2)$ and increasing in $(2,6)$. Its supremum is the maximum of $f(1)$ and $f(6)$ (which is $f(6)$) but this supremum is not attained in the ...
user avatar
  • 5,688
1 vote
Accepted

How to find global maxima in open intervals

On a closed interval, a continuously differentiable function can achieve its maximum either at a local maximum or at either of the edges of the interval. On an open interval, a continuously ...
user avatar
  • 116k
4 votes
Accepted

Minimum of a function defined by a sum

I think I have found an answer for the even case (when $n$ is even). Indeed, let $n=2p$ and since $\varphi(x) = -\varphi(2p-x)$ it is enough to study the following case: $x\in(2l-1, 2l)$ for $l\in\{1,...
user avatar
  • 5,559
2 votes

Apart from being maxima, minima or inflection point, can a critical point be anything else?

Your function $f$ isn't differentiable at $x = 1$. A point $a$ in the domain of the function $f$ is a critical point iff $f'(a) = 0$ (Some people do define a critical point to be a point where either ...
user avatar
0 votes

Minimum value of $\left|z^2-z+1\right|+\left|z^2+z+1\right|$ for $z\in \mathbb{C}$

For part 2: Let $a = z^2 + 1$ and $b = z$. We have $a - 1 = b^2$. We have \begin{align*} &(|z^2 - z + 1| + |z^2 + z + 1|)^2\\ =\,& |a - b|^2 + |a + b|^2 + 2|a - b|\cdot |a + b| \\ =\,& ...
user avatar
  • 24.8k
1 vote

The function $y=f(x)$ is represented parametrically by $x=t^5-5t^3-20t+7$ and $y=4t^3-3t^2-18t+3$ $(-2\lt t\lt2)$. The minimum of $y=f(x)$ occurs at

A short answer is that when we apply parametric first derivatives to a curve, we also have to give attention to the direction "traced" along the curve by increasing $ \ t \ \ $ (the same is ...
user avatar
  • 2,531
4 votes

minimum of $\sum\limits_{k=1}^n |1+z_k|+|1+z_1\cdots z_n|$ for $n$ even, $z\in\mathbb{C}$?

Fact 1: Let $w_1, w_2, w_3 \in \mathbb{C}$ with $|w_3| \le 1$. Then $|1 + w_1| + |1 + w_2| + |1 + w_1 w_2 w_3| \ge |1 + w_3|$. (The proof is given at the end.) Fact 2: Let $w_1, w_2, w_3 \in \mathbb{C}...
user avatar
  • 24.8k
10 votes
Accepted

minimum of $\sum\limits_{k=1}^n |1+z_k|+|1+z_1\cdots z_n|$ for $n$ even, $z\in\mathbb{C}$?

The answer is still $2$. We use induction on $n$. You've linked the basis case $(n=2)$. Now we make $2$ cases: Case 1: Modulus of at least two of the complex numbers is $1$ or more. Without loss of ...
user avatar
  • 14.4k
2 votes

Find the maximum attained by $f(x,y)=\frac{|a_1+a_2x+a_3(y-\alpha)|}{\sqrt y}$ in the triangle $\mathcal T$

Triangle $ \ \mathcal{T} \ $ has vertices $ \ A \ (0 \ , \ \alpha) \ \ , \ \ B \ (0 \ , \ \alpha + 1) \ \ , \ \ C \ (1 \ , \ \alpha + 1) \ \ , \ \ \alpha \ > \ 0 \ \ , $ with the "oblique"...
user avatar
  • 2,531
2 votes
Accepted

What are the minimum and maximum values here?

The function $f: \mathbb{R}_+ \rightarrow \mathbb{R}, x\mapsto f(x)=x+1/x$ is continuously differentiable for all $x$ in its domain. Its derivative is $$f'(x)=1-1/x^2.$$ Thus an extremum $x^*$ is ...
user avatar
  • 594
2 votes

The function $y=f(x)$ is represented parametrically by $x=t^5-5t^3-20t+7$ and $y=4t^3-3t^2-18t+3$ $(-2\lt t\lt2)$. The minimum of $y=f(x)$ occurs at

The function $y=f(x)$ is represented parametrically by $x=t^5-5t^3-20t+7$ and $y=4t^3-3t^2-18t+3$ $(-2\lt t\lt2)$. The minimum of $y=f(x)$ occurs at $t=...$ Alternative approach: Since you are (in ...
user avatar
  • 24.6k
2 votes

The function $y=f(x)$ is represented parametrically by $x=t^5-5t^3-20t+7$ and $y=4t^3-3t^2-18t+3$ $(-2\lt t\lt2)$. The minimum of $y=f(x)$ occurs at

I have plotted the parametrised function for $-2\leq t\leq 2$ and i got with $A=f\left(\frac{3}{2}\right)$, thus there is indeed a minimum. The question is why and the point is that the ...
user avatar
1 vote

The function $y=f(x)$ is represented parametrically by $x=t^5-5t^3-20t+7$ and $y=4t^3-3t^2-18t+3$ $(-2\lt t\lt2)$. The minimum of $y=f(x)$ occurs at

When I plot the parametric equation in desmos, I see a local min at approximately $x = -32$. Then, I solve the equation $t^5-5t^3-20t+7 = -32$. One of the roots is very close to $3/2$. So, I think $...
user avatar
  • 1,195
2 votes
Accepted

Find the maximum attained by $f(x,y)=\frac{|a_1+a_2x+a_3(y-\alpha)|}{\sqrt y}$ in the triangle $\mathcal T$

Let $$g(x, y) := [f(x, y)]^2 = \frac{[a_1+a_2x+a_3(y-\alpha)]^2}{y}.$$ We need to find the maximum of $g(x, y)$ subject to $0 \le x \le y - \alpha$ and $\alpha \le y \le \alpha + 1$. Fact 1: If $F(u)$ ...
user avatar
  • 24.8k
1 vote

Maximum-Minimum value of sum of modulus of complex numbers

Let $$F(u)=|1+z|+|1-z+z^2|, z=e^{iu}, u\in \Re$$ $$F(u)=2|\cos(u/2)|+\sqrt{[1-\cos u+ \cos (2u)]^2+[\sin u -\sin (2u)]^2}$$ $$\implies F(u)=2|\cos(u/2)|+\sqrt{(2 \cos u-1)^2}=2|\cos(u/2)|+|2\cos u-1|.$...
user avatar
  • 39.3k
1 vote
Accepted

Confusion regarding a continuous function achieving its minimum.

The $(a_j,b_j)$ don't cover the whole domain, they just consist of the points where $f>\lambda$. The point then is that $f$ must equal $\lambda$ at the endpoints of these intervals, otherwise ...
user avatar
  • 94.9k
1 vote

Find the maximum attained by $f(x,y)=\frac{|a_1+a_2x+a_3(y-\alpha)|}{\sqrt y}$ in the triangle $\mathcal T$

Forming the lagrangian $$ L(x,y,\lambda,s) = \frac{1}{y}(a_1+a_2 x+a_3(y-\alpha))^2+\lambda_1(x-s_1^2)+\lambda_2(x-y+\alpha+s_2^2)+\lambda_3(y-\alpha-1+s_3^2) $$ here $s_1,s_2,s_3$ are slack variables ...
user avatar
  • 26.5k
1 vote

Determine the points in $[0, π]$ at which $f(x) = e^{2 \sin x−x}$ attains its maximum and minimum values.

So we want to find the maximum and minimum values of the function $$f(x)=e^{2\sin(x)-x}.$$ A differentiable function $f$ has a maximum (or minimum) if $f'=0$. Since $e^x$ and $2\sin(x)-x$ are ...
user avatar
3 votes
Accepted

Determine the points in $[0, π]$ at which $f(x) = e^{2 \sin x−x}$ attains its maximum and minimum values.

Once you differentiate $f(x)$ and find $f'(x)=(2\cos(x)-1)e^{2\sin(x)-x}$, you should determine where $f'(x)$ is positive. The inequality $(2\cos(x)-1)e^{2\sin(x)-x}> 0$ can be re-written as $2\cos(...
user avatar
  • 393

Top 50 recent answers are included