Question 996105: Can you find out the largest and the smallest values for (a) x^2 + y^2 (b) x^2 - y^2 if -10 <= x <= 10 and -5 <= y <= 5?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! (a)
If f(x) = x^2 and the domain is restricted to [-10,10], then the range is [0,100]. 0 is the smallest output, 100 is the largest
x^2 maxes out at 100
If f(x) = x^2 and the domain is restricted to [-5,5], then the range is [0,25]. 0 is the smallest output, 25 is the largest
y^2 maxes out at 25
Overall, x^2 + y^2 will max out to 100+25 = 125
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The smallest x^2 can get is 0 (see above)
The smallest y^2 can get is 0 (see above)
The smallest x^2+y^2 can get is 0+0 = 0
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Summary:
largest value = 125
smallest value = 0
0 <= x^2+y^2 <= 125
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(b)
Look back at part (a) to find that x^2 maxes out at 100
the smallest y^2 gets is 0
x^2 - y^2 maxes out at 100-0 = 100
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The smallest x^2 gets is 0
The largest y^2 gets is 25
0-25 = -25
The smallest x^2 - y^2 gets is -25
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Summary:
largest value = 100
smallest value = -25
-25 <= x^2-y^2 <= 100
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