Let f(x) = x^3 + 8x + 3. Which one of the pro | vedclass

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(D) To determine if a function $f(x)$ has an inverse,we check if it is strictly monotonic (either strictly increasing or strictly decreasing).Given $f

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$f'(x)$ is always positive.

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(D) To determine if a function $f(x)$ has an inverse,we check if it is strictly monotonic (either strictly increasing or strictly decreasing).Given $f(x) = x^3 + 8x + 3$.The derivative is $f'(x) = \frac{d}{dx}(x^3 + 8x + 3) = 3x^2 + 8$.Since $x^2 \ge 0$ for all real $x$,it follows that $3x^2 \ge 0$.Therefore,$f'(x) = 3x^2 + 8 \ge 8$.Since $f'(x) > 0$ for all $x \in \mathbb{R}$,the function $f(x)$ is strictly increasing on its entire domain.$A$ strictly monotonic function is always one-to-one (injective) and thus possesses an inverse.Therefore,the property that $f'(x)$ is always positive allows us to conclude that $f(x)$ has an inverse.

Let $f(x) = x^3 + 8x + 3$. Which one of the properties of the derivative enables you to conclude that $f(x)$ has an inverse?

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