Is y = 5 an upper bound for f(x) = x^2 + 5?

1 Answer
Trevor Ryan.
Oct 11, 2015

No

Explanation:

By definition, a function f(x) is bounded above if there exists a real number M in RR such that the function value of f(x) < M for all x in RR
If a particular upper bound M is the lowest possible upper bound for a function, then it is called the supremum (sup).

Since f(x)=x^2+5 >=5 AA x in RR, it implies that y=5 is not an upper bound for f.
In fact, f is not bounded above at all since it diverges to infinity.

However, 5 could be considered a lower bound for f and in fact, the greatest lower bound (infimum) of f since any value bigger than 5 is no longer a lower bound.

graph{x^2+5 [-47.8, 56.25, -10, 42]}

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