Let E = x^{2017} + y^{2017} + z^{2017} - 2017 | vedclass

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Question

(B) By the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequality,for non-negative real numbers $x, y, z$,we have:
$\frac{x^{2017} + y^{2017} + z^{201

Vedclass · Accepted Answer

$-2014$

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(B) By the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequality,for non-negative real numbers $x, y, z$,we have:
$\frac{x^{2017} + y^{2017} + z^{2017}}{3} \geq \sqrt[3]{x^{2017} y^{2017} z^{2017}}$
$x^{2017} + y^{2017} + z^{2017} \geq 3(xyz)^{2017/3}$.
However,this does not directly lead to the expression $2017xyz$.
Consider the case where $x = y = z = k$. Then $E = 3k^{2017} - 2017k^3$.
If $x=y=z=1$,then $E = 1 + 1 + 1 - 2017 = -2014$.
If $x=y=z=0$,then $E = 0$.
Since $x, y, z \geq 0$,we can test values. If we set $y=z=0$,then $E = x^{2017} \geq 0$.
If we set $x=y=z=1$,$E = -2014$.
If we consider the function $f(x, y, z) = x^{2017} + y^{2017} + z^{2017} - 2017xyz$,for $x, y, z \geq 0$,the minimum value occurs at $x=y=z=1$,which gives $1+1+1-2017 = -2014$.

Let $E = x^{2017} + y^{2017} + z^{2017} - 2017xyz$ (where $x, y, z \geq 0$),then the least value of $E$ is

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