question\ngiven the graph of $f(x)$ below, find the graph of the derivative of $f(x)$.\nselect the correct…

Explanation:

Step1: Identify the vertex of the parabola

The vertex of $f(x)$ is at $x = 0$, where the slope is zero. $$f'(0) = 0$$

Step2: Determine the sign of the derivative

For $x < 0$, $f(x)$ is decreasing, so $f'(x) < 0$. For $x > 0$, $f(x)$ is increasing, so $f'(x) > 0$. $$f'(x) < 0 \text{ for } x < 0, f'(x) > 0 \text{ for } x > 0$$

Step3: Calculate the slope of the derivative

The function $f(x)$ passes through $(0, -1)$ and $(2, 3)$. Assuming $f(x) = ax^2 + c$, then $c = -1$ and $3 = a(2)^2 - 1$, so $a = 1$. $$f(x) = x^2 - 1 \implies f'(x) = 2x$$

Step4: Match with the correct graph

The derivative $f'(x) = 2x$ is a line passing through $(0,0)$ with a positive slope of $2$. $$\text{Graph passes through } (0,0), (1,2), (2,4)$$

Answer:

The second option (the graph of a line passing through the origin $(0,0)$ with a positive slope).