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Range of a function

Question: Find the range of the function given by . Solution: The domain of the given function is closed interval . Since the function is continuous on a closed interval, therefore it attains its bound (the greatest and least value), somewhere in the closed interval. By Fermat’s theorem if the function attains its minimum or maximum […]

Question: Find the range of the function f given by f(x)=x−−√+1−x−−−−−√.

Solution: The domain of the given function is closed interval [0,1]. Since the function f is continuous on a closed interval,

therefore it attains its bound (the greatest and least value), somewhere in the closed interval.

By Fermat’s theorem if the function attains its minimum or maximum at an interior point then the derivative at that point should either vanish or should be non-existant.

Now f′(x)=12x√−121−x√. Equating f′(x) to zero, we get x=12
Thus f(0)=1,f(1)=1,f(12)=2–√, and therefore the range of the function is closed interval [1,2–√]

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