1. The problem states that $f: \mathbb{R} \to \mathbb{R}$ is an increasing function for all $x \in \mathbb{R}$. We need to determine which graph represents $f'(x)$, the derivative of $f$. 2. Recall that if a function $f$ is increasing on an interval, then its derivative $f'(x)$ is non-negative on that interval. This means $f'(x) \geq 0$ for all $x$. 3. Let's analyze the given graph options: - (a) A downward sloping line crossing the y-axis at positive $y$ means $f'(x)$ decreases and becomes negative for large $x$. This contradicts $f$ being increasing everywhere. - (b) An upward sloping line crossing the y-axis at negative $y$ means $f'(x)$ starts negative and increases, so $f'(x)$ is negative for some $x$. This contradicts $f$ being increasing everywhere. - (c) A parabola opening upwards with vertex at origin means $f'(x) \geq 0$ for all $x$ since the parabola is always above or on the $x$-axis. This matches the condition for $f$ increasing everywhere. - (d) A cubic-like curve crossing below the $x$-axis means $f'(x)$ is negative for some $x$, contradicting $f$ increasing everywhere. 4. Therefore, the correct graph representing $f'(x)$ for an increasing function $f$ is (c), the parabola opening upwards with vertex at the origin. Final answer: The figure representing $f'(x)$ is (c) a parabola opening upwards with vertex at origin.