Answer :
The absolute maximum and minimum of the function f(x,y)=x²+14x+y²+20y+7 in the given domain −1≤x≤1,−2≤y≤2 are 66 at (1, 2) and -42 at (−1,−2) respectively. Correct option is d).
To find the absolute maxima and minima of a function on a given domain, calculate the function value at all possible points.
Then, compare these values to determine which is the highest (maximum) and lowest (minimum).
In this case, the function is f(x,y)=x²+14x+y²+20y+7, and the domain is −1≤x≤1,−2≤y≤2.
Calculate the function value at these extremes and corners, that is, at (−1,−2), (−1,2), (1,−2), and (1,2).
Out of these, the highest value is the maximum and the lowest is the minimum.
In conclusion, the absolute maximum is 66 found at (1, 2) and absolute minimum is -42 at (−1,−2).
Correct option is d).
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Correct question:
Find the absolute maxima and minima of the function on the given domain. f(x,y)=x²+14x+y²+20y+7 on the rectangular region −1≤x≤1,−2≤y≤2
a) absolute maximum 63 at (2,2) absolute minimum -42 at (−1,−2)
b) Absolute maximum: 66 at (1,2), absolute minimum: 7 at (0,0)
c) None of the other choices
d) Absolute maximum 66 at (1, 2) absolute minimum -42 at (−1,−2)
e) Absolute maximum, 66 at (1.2) absolute minimum: -117 at (−2,−10)
e) Absolute maximum: 83 at (2.2); absolute minimum: 7 at (0,0)
