SOLUTION: Find the maximum value of the function f(x) = x/2 + 2/x in interval [1, 6]?

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Question 1184290: Find the maximum value of the function f(x) = x/2 + 2/x in interval [1, 6]?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+x%2F2+%2B+2%2Fx ===> %22f%27%28x%29%22+=+1%2F2+-+2%2Fx%5E2. Letting %22f%27%28x%29%22+=+0 ===> x = 2. (x ∈ [1,6]).

%22f%27%27%28x%29%22+=+4%2Fx%5E3 and %22f%27%27%282%29%22+=+4%2F2%5E3+=+1%2F2+%3E+0 ===> there is local minimum at x = 2.

Therefore absolute max exists at either interval endpoint.

f(1) = 2.5 and f(6) = 10/3.

Hence max value of f(x) is 10/3, which can be found at x = 6.
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Alternatively, knowing that a+%2B+b+%3E=+2sqrt%28ab%29 for a,b > 0

===> x%2F2+%2B+2%2Fx+%3E=+2sqrt%28%28x%2F2%29%282%2Fx%29%29+=+2, with equality happening only for x = 2.
Hence there is minimum of 2 at x = 2. Absolute max then exists at either interval endpoint of [1,6],
and we get the same conclusion as in previous calculus approach.