1. **State the problem:** We are given a function $f(x)$ with roots at $x = -2.5$, $x = -0.75$, and $x = 0.75$, a peak at $(0, 2)$, and a minimum at approximately $(-1.9, -5.7)$. We need to determine which statement about the intervals where $f(x)$ is positive or negative is true. 2. **Analyze the roots and behavior:** The roots divide the $x$-axis into four intervals: $(-\infty, -2.5)$, $(-2.5, -0.75)$, $(-0.75, 0.75)$, and $(0.75, \infty)$. 3. **Use the given points and shape:** - At $x=0$, $f(0) = 2 > 0$, so on $(-0.75, 0.75)$ the function is positive. - The minimum at $(-1.9, -5.7)$ is negative, so on $(-2.5, -0.75)$ the function dips below zero. - The function crosses the $x$-axis at $-2.5$ and $-0.75$, so it changes sign at these points. - For $x > 0.75$, the function passes through zero and then decreases, so $f(x) < 0$ on $(0.75, \infty)$. - For $x < -2.5$, the function is above the $x$-axis since it is increasing towards the minimum at $-1.9$. 4. **Summarize signs on intervals:** - $(-\infty, -2.5)$: $f(x) > 0$ - $(-2.5, -0.75)$: $f(x) < 0$ - $(-0.75, 0.75)$: $f(x) > 0$ - $(0.75, \infty)$: $f(x) < 0$ 5. **Check the statements:** - Statement 1: $f(x) > 0$ over $(-2.5, -0.75)$ and $(0.75, \infty)$ — False, these intervals are negative. - Statement 2: $f(x) < 0$ over $(-2.5, -0.75)$ and $(-0.75, \infty)$ — False, $(-0.75, 0.75)$ is positive. - Statement 3: $f(x) < 0$ over $(-\infty, -2.5)$ and $(-0.75, 0.75)$ — False, $(-\infty, -2.5)$ and $(-0.75, 0.75)$ are positive. - Statement 4: $f(x) > 0$ over $(-\infty, -2.5)$ and $(-0.75, 0.75)$ — True. **Final answer:** Statement 4 is true.