Articles of maxima minima

if $x+y+z=1$ then find the maximum of the form

Let $x>0$, $y>0$ and $z>0$ such that $x+y+z=1$. Find the maximal value of $$x\sqrt{y}+y\sqrt{z}$$ I think $x=y=\dfrac{4}{9},z=\dfrac{1}{9}$ then the maximum $\dfrac{4}{9}$,but how to use AM-GM prove it?

Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$

Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$ ($a,b,c$ are positive real numbers). There is a solution, which relies on guessing the minimum case happening at $a=b=c=\frac12$ and then applying AM-GM inequality,but what if one CANNOT guess that?!

Show $g(x)=\frac{x \sin x}{x+1}$ has no maxima in $(0,\infty)$

We are given $g(x)=\frac{x \sin x}{x+1}$, and as I said we need to show it has no maxima in $(0,\infty)$. My attempt: assume there is some $x_0>0$ that yields a maxima. then for all $x$ $$-1+\frac{1}{x+1}\leq \frac{x \sin x}{x+1}\leq \frac{x_0 \sin x_0}{x_0+1}\leq 1-\frac{1}{x_0+1}$$ and we can find some $x$ for which this isn’t satisfied (like […]

A smooth function $f(x)$ has a unique minimum. It $f$ also varies smoothly in time, does the location of its minimum vary smoothly in time?

Let $f(x,t)$ be a smooth function $\mathbb R^2\to\mathbb R$ such that $F_t(x):=f(x,t)$ has a unique minimum in $x$ for every fixed $t\in[0,1]$. How regularly does the location of this unique minimum vary with respect to $t$? In other words, if $x=\chi(t)$ is the $x$-value where $F_t(x)$ attains its unique minimum, can we say that $\chi(t)$ […]

Maximum value of expression: $\frac{ab+bc+cd}{a^2+b^2+c^2+d^2}$

What is the maximum value of $$\frac{ab+bc+cd}{a^2+b^2+c^2+d^2}$$ where $a,b,c$, and $d$ are real numbers?

Find the minimum value of $P=\sum _{cyc}\frac{\left(x+1\right)^2\left(y+1\right)^2}{z^2+1}$

For $x>0$, $y>0$, $z>0$ and $x+y+z=3$ find the minimize value of $$P=\frac{\left(x+1\right)^2\left(y+1\right)^2}{z^2+1}+\frac{\left(y+1\right)^2\left(z+1\right)^2}{x^2+1}+\frac{\left(z+1\right)^2\left(x+1\right)^2}{y^2+1}$$ We have: $P=\left(\left(x+1\right)\left(y+1\right)\left(z+1\right)\right)^2\left(\frac{1}{\left(z+1\right)^2\left(z^2+1\right)}+\frac{1}{\left(y+1\right)^2\left(y^2+1\right)}+\frac{1}{\left(x+1\right)^2\left(x^2+1\right)}\right)$ $\ge \left(\left(x+1\right)\left(y+1\right)\left(z+1\right)\right)^2\left(\frac{1}{2\left(z^2+1\right)^2}+\frac{1}{2\left(x^2+1\right)^2}+\frac{1}{2\left(y^2+1\right)^2}\right)$ $\ge \left(\left(x+1\right)\left(y+1\right)\left(z+1\right)\right)^2\left(\frac{9}{2\left(\left(z^2+1\right)^2+\left(y^2+1\right)^2+\left(x^2+1\right)^2\right)}\right)$ I can’t continue. Help

a two-variable cyclic power inequality $x^y+y^x>1$ intractable by standard calculus techniques

This question already has an answer here: Exponential teaser [closed] 1 answer

Find max: $\frac{a}{b+2a}+\frac{b}{c+2b}+\frac{c}{a+2c}$

This question already has an answer here: Olympiad inequality $\frac{a}{2a + b} + \frac{b}{2b + c} + \frac{c}{2c + a} \leq 1$. 2 answers

Calculus of variations ( interpreting the minimum in first order)

I have just begun studying the Calculus of Variations and I have 3 doubts in it. I have written certain things in bold so as anyone who wishes to answer the question but find it quite long can just go through the bold part and he would get an idea as what my doubt is […]

How to find $\max\int_{a}^{b}\left (\frac{3}{4}-x-x^2 \right )\,dx$ over all possible values of $a$ and $b$, $(a<b)$?

I tried finding the maxima of $f(x)=\frac{3}{4}-x-x^2$ by taking the derivative and so on and use the fact that $\displaystyle\int_{a}^{b}f(x)\,dx \leq M(b-a)$ where $M$ is the global maximum, but then the maximum value depends on the values of $a$ and $b$.