determine the absolute minimum value of (m = x - y) subject to the constraint (y=\frac{x^{4}+1024}{256})…

Explanation:

Step1: Substitute the constraint

Substitute $y = \frac{x^{4}+ 1024}{256}$ into $M = x - y$. So $M=x-\frac{x^{4}+1024}{256}$.

Step2: Differentiate $M$ with respect to $x$

Using the power - rule, if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. The derivative of $M$ with respect to $x$ is $M^\prime=1-\frac{4x^{3}}{256}=1-\frac{x^{3}}{64}$.

Step3: Set the derivative equal to zero

Set $M^\prime = 0$, so $1-\frac{x^{3}}{64}=0$. Then $\frac{x^{3}}{64}=1$, and $x^{3}=64$, which gives $x = 4$.

Step4: Find the second - derivative of $M$

Differentiate $M^\prime=1-\frac{x^{3}}{64}$ with respect to $x$. $M^{\prime\prime}=-\frac{3x^{2}}{64}$.

Step5: Evaluate the second - derivative at $x = 4$

Substitute $x = 4$ into $M^{\prime\prime}$, $M^{\prime\prime}(4)=-\frac{3\times4^{2}}{64}=-\frac{3\times16}{64}=-\frac{3}{4}<0$. So $x = 4$ is a local maximum.

Step6: Substitute $x = 4$ into $M$

First, find $y$ when $x = 4$. $y=\frac{4^{4}+1024}{256}=\frac{256 + 1024}{256}=\frac{1280}{256}=5$. Then $M=x - y$, so $M=4 - 5=-1$.

Answer:

$-1$