Questions: Given the function (f(x)=x(2 cos x^2-2)), it is true that Select one: (f) is bounded from above (f) is unbounded from above and has no limit as (x rightarrow+infty) (lim x rightarrow+infty f(x)=0) (f) is bounded from below

Given the function (f(x)=x(2 cos x^2-2)), it is true that
Select one:
(f) is bounded from above
(f) is unbounded from above and has no limit as (x rightarrow+infty)
(lim x rightarrow+infty f(x)=0)
(f) is bounded from below

Transcript text: Given the function $f(x)=x\left(2 \cos x^{2}-2\right)$, it is true that Select one: $f$ is bounded from above $f$ is unbounded from above and has no limit as $x \rightarrow+\infty$ $\lim _{x \rightarrow+\infty} f(x)=0$ $f$ is bounded from below

Solution

Solution Steps

Step 1: Analyze the function $ f(x) = x(2 \cos x^2 - 2) $

The function $ f(x) $ is a product of $ x $ and $ (2 \cos x^2 - 2) $. The term $ \cos x^2 $ oscillates between $-1$ and $1$ as $ x \rightarrow +\infty $, so $ 2 \cos x^2 - 2 $ oscillates between $-4$ and $0$.

Step 2: Determine the behavior of $ f(x) $ as $ x \rightarrow +\infty $

Since $ x $ grows without bound as $ x \rightarrow +\infty $, and $ 2 \cos x^2 - 2 $ oscillates between $-4$ and $0$, the product $ f(x) = x(2 \cos x^2 - 2) $ will oscillate between $-\infty$ and $0$. This means $ f(x) $ is unbounded from below and does not approach a finite limit as $ x \rightarrow +\infty $.

Step 3: Evaluate the given options

$ f $ is bounded from above: False, because $ f(x) $ oscillates and does not have an upper bound.
$ f $ is unbounded from above and has no limit as $ x \rightarrow +\infty $: True, as $ f(x) $ oscillates and does not approach a finite limit.
$\lim _{x \rightarrow +\infty} f(x) = 0$: False, because $ f(x) $ does not approach a finite limit.
$ f $ is bounded from below: False, because $ f(x) $ is unbounded from below.
$\lim _{x \rightarrow +\infty} f(x) = +\infty$: False, because $ f(x) $ does not approach $ +\infty $.

Final Answer

The correct answer is B.

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