anemone said:If $x^2+y^2+{\color{red}z}^2+xyz=4$ and that $x,\,y,\,x\ge 0$, prove $3!\sqrt{x}+2!y+1!z^2\le 13$.
Olinguito said:Is there a typo? Should it be …
Olinguito said:My attempt.
We have (by AM–GM or otherwise):
$$\begin{array}{rcl}6\sqrt x &\le& x^2+\dfrac{81}{16}+1+1 \\ 2y &\le& y^2+1 \\ z^2 &\le& z^2+xyz\end{array}$$
$\begin{array}{rl}\implies & 6\sqrt x+2y+z^2 \\ \le & x^2+y^2+z^2+xyz+\dfrac{129}{16}\ =\ \dfrac{193}{16}\ <\ 13.\end{array}$
Olinguito said:$$\frac{x^2+\dfrac{81}{16}+1+1}4\ \ge\ \sqrt[4]{(x^2)\left(\frac{81}{16}\right)(1)(1)}$$
(AM–GM). Did I get it wrong? (Sweating)
The inequality expression 3√{x}+2y+1z^2⩽ 13 represents a mathematical statement that compares the values of three variables, x, y, and z, to the number 13. The expression includes a square root, a cube root, and a square term, making it a radical, square, and factorial expression.
The radical, square, and factorial terms in this inequality expression indicate that the values of the variables are being manipulated in different ways. The radical term (√{x}) represents the square root of x, the square term (z^2) represents z multiplied by itself, and the factorial term (y!) represents the product of all the positive integers from 1 to y.
The values of x, y, and z that satisfy the inequality 3√{x}+2y+1z^2⩽ 13 are any values that, when substituted into the expression, result in a value less than or equal to 13. This could include a range of values for each variable, depending on the specific values chosen.
This inequality expression can be graphed on a three-dimensional coordinate plane, with the x-axis representing the values of x, the y-axis representing the values of y, and the z-axis representing the values of z. The graph would show all the points in space that satisfy the inequality, creating a three-dimensional surface.
This inequality expression could represent a variety of real-world scenarios, such as budget constraints, physical limitations, or resource allocations. For example, it could represent the maximum amount of money (represented by the number 13) that can be spent on three different expenses (represented by the variables x, y, and z) while staying within certain constraints (represented by the radical, square, and factorial terms).