Is there an absolute maximum value of this function?

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SUMMARY

The function f: R² -> R defined by f(x,y) = e^(x+y) - y + x is analyzed for its absolute maximum on the set s = {(x,y) : |x| + |y| <= 2}. The discussion confirms that the set s is both bounded and closed, thus compact. By applying the Extreme Value Theorem (EVT), it is established that f has an absolute maximum on s. The challenge arises in determining the boundary of s, which can be approached by considering the geometric representation of the set as a diamond shape in the Cartesian plane.

PREREQUISITES
  • Understanding of continuous functions in multivariable calculus
  • Familiarity with the Extreme Value Theorem (EVT)
  • Knowledge of compact sets in topology
  • Basic skills in geometric interpretation of inequalities
NEXT STEPS
  • Study the properties of continuous functions on compact sets
  • Learn about the geometric representation of inequalities in R²
  • Explore examples of applying the Extreme Value Theorem in multivariable contexts
  • Investigate methods for finding boundaries of sets defined by absolute values
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Students and educators in calculus, particularly those focusing on multivariable functions and optimization problems, as well as mathematicians interested in the application of the Extreme Value Theorem.

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Homework Statement



Consider the function f:R^2->R defined by f(x,y)=[e^(x+y)]-y+x. Is there an absolute maximum value of f on the set s={(x,y):/x/+/y/<=2}? Justify.

note, /x/ is the absolute value of x.

Homework Equations



a. If f is con't, it takes compact sets to compact sets.

b.Extreme value thm: Suppose s belongs to R^n is compact and f: s->R is continuous. then f has an absolute min value and an absolute max value on S


The Attempt at a Solution




My idea is
=>show s is bounded and closed => therefore compact => f con't maps compact sets to compact sets => EVT

but I encounted problem when finding boundary of s
/x/+/y/<=2 that means /x/<=2 when y=0 and /y/<=2 when x=0. then can I find the boundary here by constructing a circle with x and y? because I saw other example did it on this way.
Help..
 
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