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

Explanation:

Step1: Substitute the constraint into the function

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 $M(x)=x-\frac{1}{256}x^{4}-\frac{1024}{256}$, then $M^\prime(x)=1-\frac{4}{256}x^{3}=1 - \frac{1}{64}x^{3}$.

Step3: Set the derivative equal to zero to find critical points

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

Step4: Find the second - derivative of $M$

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

Step5: Evaluate the second - derivative at the critical point

Substitute $x = 4$ into $M^{\prime\prime}(x)$. $M^{\prime\prime}(4)=-\frac{3}{64}\times16=-\frac{3}{4}<0$, which means the function has a local maximum at $x = 4$.

Step6: Substitute $x = 4$ into the original function for $M$

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:

$-1$