Inequality involves radical, square and factorial expression 3√{x}+2y+1z^2⩽ 13

In summary, the conversation discusses a mathematical expression involving variables x, y, and z, and the result being less than or equal to 13. The symbols √, +, and ^2 represent square root, addition, and squaring, respectively. In order to solve this expression for a specific value of x, y, or z, more information is needed. The expression can be graphed in 3D and has potential real-world applications in fields such as physics, engineering, and finance.
  • #1
anemone
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If $x^2+y^2+z^2+xyz=4$ and that $x,\,y,\,x\ge 0$, prove $3!\sqrt{x}+2!y+1!z^2\le 13$.
 
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  • #2
Is there a typo? Should it be …

anemone said:
If $x^2+y^2+{\color{red}z}^2+xyz=4$ and that $x,\,y,\,x\ge 0$, prove $3!\sqrt{x}+2!y+1!z^2\le 13$.
 
  • #3
Olinguito said:
Is there a typo? Should it be …

Indeed it is a typo(Tmi), sorry about that, and I just fixed it! Thanks for catching the typo!(Yes)
 
  • #4
My attempt.

We have (by AM–GM or otherwise):
$$\begin{array}{rcl}6\sqrt x &\le& x^2+\dfrac{81}{16}+1+1 \\ 2y &\le& y^2+1 \\ z^2 &\le& z^2+xyz\end{array}$$
$\begin{array}{rl}\implies & 6\sqrt x+2y+z^2 \\ \le & x^2+y^2+z^2+xyz+\dfrac{129}{16}\ =\ \dfrac{193}{16}\ <\ 13.\end{array}$
 
  • #5
Olinguito said:
My attempt.

We have (by AM–GM or otherwise):
$$\begin{array}{rcl}6\sqrt x &\le& x^2+\dfrac{81}{16}+1+1 \\ 2y &\le& y^2+1 \\ z^2 &\le& z^2+xyz\end{array}$$
$\begin{array}{rl}\implies & 6\sqrt x+2y+z^2 \\ \le & x^2+y^2+z^2+xyz+\dfrac{129}{16}\ =\ \dfrac{193}{16}\ <\ 13.\end{array}$

I do not understand the 1st line
 
  • #6
$$\frac{x^2+\dfrac{81}{16}+1+1}4\ \ge\ \sqrt[4]{(x^2)\left(\frac{81}{16}\right)(1)(1)}$$
(AM–GM). Did I get it wrong? (Sweating)
 
  • #7
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)

I never said it was wrong. I said In did not understand. Thanks or clarifying. It is correct
 

Related to Inequality involves radical, square and factorial expression 3√{x}+2y+1z^2⩽ 13

What is the meaning of the inequality expression 3√{x}+2y+1z^2⩽ 13?

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.

What is the significance of the radical, square, and factorial terms in this inequality 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.

What values of x, y, and z satisfy the inequality 3√{x}+2y+1z^2⩽ 13?

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.

How can this inequality expression be graphed?

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.

What real-world scenarios could be represented by this inequality expression?

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).

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