Maximum Value of $x$ with Given Constraints?

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In summary, the maximum of x refers to the highest value that x can reach in a set of data or in a mathematical equation. To find the maximum of x in a set of data, the data can be arranged in ascending order and the last value in the list will be the maximum value of x. Calculus is used to find the maximum of x in a continuous function or curve by taking the derivative and evaluating the critical points. The maximum of x can change in a data set if new data points are added or if existing data points are modified. The maximum of x is different from the minimum of x, as the maximum is the upper limit and the minimum is the lower limit for x in a given set of data or equation.
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anemone
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Let $x,\,y,\,z$ be real numbers such that $x+2y+3z=86$ and $x^2+y^2+z^2=2014$.

Find the maximum of $x$.
 
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My Solution:

Given $2y+3z = 86-x$ and $y^2+z^2 = 2014-x^2$

Now Using Cauchy-Schwartz Inequality, We Get

$\displaystyle (2^2+3^2)\cdot (y^2+z^2)\geq (2y+3z)^2\Rightarrow 13\cdot (2014-x^2)\geq (86-x)^2$

So $\displaystyle (7x-303)\cdot (x+31)\leq 0$

So $\displaystyle -31\leq x \leq \frac{303}{7}$
 
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jacks said:
My Solution:

Given $2y+3z = 86-x$ and $y^2+z^2 = 2014-x^2$

Now Using Cauchy-Schwartz Inequality, We Get

$\displaystyle (2^2+3^2)\cdot (y^2+z^2)\geq (2y+3z)^2\Rightarrow 13\cdot (2014-x^2)\geq (86-x)^2$

So $\displaystyle (7x-303)\cdot (x+31)\leq 0$

So $\displaystyle -31\leq x \leq \frac{303}{7}$

Bravo, jacks!(Yes) And thanks for participating!:)
 

FAQ: Maximum Value of $x$ with Given Constraints?

What is the meaning of "maximum of x"?

The maximum of x refers to the highest value that x can reach in a given set of data or in a mathematical equation. It is the largest possible value that x can take on.

How do you find the maximum of x in a set of data?

To find the maximum value of x in a set of data, you can arrange the data in ascending order and then select the last value in the list. This will be the maximum value of x in the data set.

What is the role of calculus in finding the maximum of x?

Calculus is used to find the maximum of x in a continuous function or curve. This is done by taking the derivative of the function and setting it equal to zero to find the critical points. The critical points are then evaluated to determine the maximum value of x.

Can the maximum of x change in a given set of data?

Yes, the maximum of x can change in a given set of data if new data points are added or if existing data points are modified. The maximum value of x is dependent on the data set and can change as the data changes.

How is the maximum of x different from the minimum of x?

The maximum of x is the highest value that x can reach, while the minimum of x is the lowest value that x can reach. In other words, the maximum value is the upper limit and the minimum value is the lower limit for x in a given set of data or equation.

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