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anemone
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Determine the largest number $k$ such that system $a^2+b^2=1$, $|a^3-b^3|+|a-b|=k^3$ has a solution.
anemone said:Determine the largest number $k$ such that system $a^2+b^2=1$, $|a^3-b^3|+|a-b|=k^3---(1)$ has a solution.
Albert said:from (1) we know that :$k\geq 0$
if $a=b$ then $k=0$
if $a>b$, we have :
$(a-b)(ab+2)=k^3=f(a,b)---(*)$
using lagrange-method:
we want to find:
$(a-b)(ab+2)+L(a^2+b^2-1)=max(f(a,b))$
we get $L=\dfrac {b-a}{2}=\dfrac {4ab+3}{2(b-a)}$
$\therefore ab=\dfrac {-1}{3}---(2)$
$a^2+b^2=1---(3)$
from (2)(3)$a-b=\sqrt{\dfrac {5}{3}}---(4)$
put (2)(4) to (*) and we get:
$k^3=\dfrac{5}{3}\times\sqrt{\dfrac{5}{3}}$
$k=\sqrt{\dfrac{5}{3}}$
if $b>a$
because of symmetry it will be the same
anemone said:Thanks for participating and well done, Albert! Your answer is correct! Something tells me you really like to use the AM-GM to solve for problem such as this one and for your information, the solution that I have used the AM-GM to solve it as well...do you think you want to give AM-GM a try? Hehehe...
Albert said:use AM-GM
$(a-b)^2=1-2ab$
$k^3=(a-b)(ab+2)=\sqrt {1-2ab}\times (ab+2)---(*)\leq\dfrac{(1-2ab)+(ab+2)^2}{2}$
$\therefore 1-2ab=ab+2,$ or $ab=\dfrac {-1}{3}$
put $ab=\dfrac {-1}{3} $to (*)we get:
$k=\sqrt {\dfrac {5}{3}}$
The largest number refers to the greatest numerical value within a given set of numbers. It is the number that is the greatest in magnitude or quantity.
To determine the largest number from a set of numbers, you can arrange the numbers in ascending order and the last number in the arrangement will be the largest number. Alternatively, you can use a calculator or a computer program to find the largest number.
The number k is usually used as a placeholder or variable in mathematical equations to represent an unknown number. In the context of determining the largest number, k can be used to represent the largest number within a given set of numbers.
Yes, the largest number can be negative. In a set of numbers that includes both positive and negative numbers, the largest number will be the one with the greatest absolute value. For example, in the set {-5, 3, -10, 7}, the largest number is -10.
It depends on the system or context in which the numbers are being used. In the decimal system, there is no limit to the largest number that can be represented. However, in certain mathematical systems such as the binary system, there is a limit to the largest number that can be represented due to the finite number of digits available.