Local extrema of functions of several variables

Investigate the extrema of the following function of several variables: $u=x+\dfrac {y^{2}}{4x}+\dfrac {z^{2}}{y}+\dfrac {2}{z}$ $(x>0, y>0, z>0)$.

Investigate the extrema of the following function of several variables: $u=xy^{2}z^{3}(a-x-2y-3z)$ $(a>0)$.

Investigate the extrema of the following function of several variables: $u=\dfrac {a^{2}}{x}+\dfrac {x^{2}}{y}+\dfrac {y^{2}}{z}+\dfrac {z^{2}}{b}$ $(x>0, y>0, z>0, a>0, b>0)$.

Investigate the extrema of the following function of several variables:

$(0 \leqslant x \leqslant \pi ; 0 \leqslant y \leqslant \pi ; 0 \leqslant z \leqslant \pi )$.

Investigate the extrema of the following function of several variables:

$(x_{1}>0, x_{2}>0, \ldots , x_{n}>0)$.

Investigate the extrema of the following function of several variables:

$(x_{i}>0, i=1,2,\ldots ,n)$.

Between two positive numbers $a$ and $b$, insert $n$ numbers $x_{1}, x_{2}, \ldots , x_{n}$ so that the fraction

attains its maximum value.

Find the extrema of the implicit function $z$ of the variables $x$ and $y$: $x^{2}+y^{2}+z^{2}-2x+2y-4z-10=0$.

Find the extrema of the implicit function $z$ of the variables $x$ and $y$: $x^{2}+y^{2}+z^{2}-xz-yz+2x+2y+2z-2=0$.

Find the extrema of the implicit function $z$ of the variables $x$ and $y$: $(x^{2}+y^{2}+z^{2})^{2}=a^{2}(x^{2}+y^{2}-z^{2})$.

Prove that the function $z=(1+e^{y})\cos x-ye^{y}$ has infinitely many local maxima and not a single local minimum.

Is it a sufficient condition for the function $f(x,y)$ to have a local minimum at the point $M_{0}(x_{0},y_{0})$ that this function have a local minimum along every straight line through $M_{0}$?

In the plane there are given $n$ mass points $P_{1}(x_{1},y_{1})$, $P_{2}(x_{2},y_{2})$, $\ldots$, $P_{n}(x_{n},y_{n})$, with masses $m_{1}$, $m_{2}$, $\ldots$, $m_{n}$ respectively. At what position of the point $P(x,y)$ is the moment of inertia of this system with respect to that point smallest?

On the interval $1 \leqslant x \leqslant 3$, approximate the function $f(x)=x^{2}$ by the linear function $a+bx$ so that

Given the condition that the mean square deviation between the functions $a+bx$ and $\sqrt{1+x^{2}}$ on the given interval $\left[0,1\right]$ is minimal, find the approximate formula

Find the extreme values of the function given explicitly: $z = x^{2}+xy+y^{2}-3ax-3by$.

Find the extreme values of the function given explicitly: $z = x^{3}y^{2}(a-x-y)$.

Find the extreme values of the function given explicitly: $z = x^{4}+y^{4}-2x^{2}+4xy-2y^{2}$.

Find the extreme values of the function given explicitly: $z = x^{2}-xy+y^{2}+3x-2y+1$.

Find the extreme values of the function given explicitly: $z = x^{2}+xy+y^{2}+\dfrac {a^{3}}{x}+\dfrac {a^{3}}{y}$, $x > 0$, $y > 0$.

Find the extreme values of the function given explicitly: $z = x^{3}+y^{3}-3axy$.

Find the extreme values of the function given explicitly: $z = Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F$.

Find the extreme values of the function given explicitly: $z = e^{-x^{2}-y^{2}}(ax^{2}+by^{2})$; $a > 0$, $b > 0$.

Find the extreme values of the function given explicitly: $z = \sqrt{(a-x)(a-y)(x+y-a)}$.

Find the extreme values of the function given explicitly: $z = \dfrac {a+bx+cy}{\sqrt{1+x^{2}+y^{2}}}$.

Find the extreme values of the function given explicitly: $z = (a\cos x+b\cos y)^{2}+(a\sin x+b\sin y)^{2}$.

Find the extreme values of the function given explicitly: $z = \cos x \cos \alpha +\sin x \sin \alpha \cos (y-\beta )$.

Find the extreme values of the function given explicitly: $u = xyz(4a-x-y-z)$.

Find the extreme values of the function given explicitly: $u = x^{2}+y^{2}+z^{2}-xy+2z+x$.

Find the extreme values of the function given explicitly: $u = \dfrac {x^{3}+y^{3}+z^{3}}{xyz}$; $x > 0$, $y > 0$, $z > 0$.

Find the extreme values of the function given explicitly: $u = \dfrac {x}{y+z}+\dfrac {y}{x+z}+\dfrac {z}{x+y}$; $x > 0$, $y > 0$, $z > 0$.

Find the extreme values of the function given explicitly: $u = (ax+by+cz)e^{-x^{2}-y^{2}-z^{2}}$.

Between the positive numbers $a$ and $b$, insert $n$ other numbers $x_{1}, x_{2}, \ldots , x_{n}$ so that the value of the fraction

attains its greatest value.

Find the greatest and least values of the function $z$, given implicitly as a function of $x$ and $y$ by the equation $2x^{2}+2y^{2}+z^{2}+8xz-z+8 = 0$.

Find the greatest and least values of the function $z$, given implicitly as a function of $x$ and $y$ by the equation $2x^{2}+6y^{2}+2z^{2}+8xz-4x-8y+3 = 0$.

Find the greatest and least values of the function $z$, given implicitly as a function of $x$ and $y$ by the equation $6x^{2}+6y^{2}+6z^{2}+4x-8y-8z+5 = 0$.

Find the greatest and least values of the function $z$, given implicitly as a function of $x$ and $y$ by the equation $5x^{2}+5y^{2}+5z^{2}-2xy-2xz-2yz-72 = 0$.

Find the greatest and least values of the function $z$, given implicitly as a function of $x$ and $y$ by the equation $x^{3}y-3xy^{2}+6x+y^{2}+7y+z^{2}-3z-14 = 0$.

Find the greatest and least values of the function $z$, given implicitly as a function of $x$ and $y$ by the equation $x^{4}+y^{4}+z^{4} = 2a^{2}(x^{2}+y^{2}+z^{2})$.

Show that there exists a function $f$ of two variables for which the equations $\partial f(x, y) / \partial x=$ 0 and $\partial f(x, y) / \partial y=0$ have a unique solution, and this solution is a local but not a global maximum of $f$.