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(Extreme Value Theorem with boundedness) If g(x) is continuous on [0,1] then g attains its maximum and it is bounded where there is a number M which is greater than or equal to 0 such that: the absolute value of g(x) is less than or equal to M in the interval [0,1]. Question: How to find a number M so that the absolute value of g(x) is less than or equal to M, where g(x) is given by g(x)= x(1-x)? Thank you.

(Extreme Value Theorem with boundedness) 
If g(x) is continuous on [0,1] then g attains its maximum and it is bounded where there is a number M which is greater than or equal to 0 such that: the absolute value of g(x) is less than or equal to M in the interval [0,1]. 
 
Question: How to find a number M so that the absolute value of g(x) is less than or equal to M, where g(x) is given by g(x)= x(1-x)?
 
Thank you. 

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Grade:12th pass

1 Answers

Aditya Gupta
2075 Points
2 years ago
x(1 – x)= g(x)
now g(0)= g(1)= 0
now x(1 – x)= x – x^2= ¼ – (x – ½)^2
now obviously (x – ½)^2 is greater than or equal to zero. for g(x) to achieve max value (x – ½)^2 should be minimum, which is zero and occurs when x= ½ .
so M= ¼ – 0 = 1/4

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