1. **State the problem:** We need to find a quartic function $f(x)$ that produces a W-shaped curve opening upward with x-intercepts at approximately $x=-2$ and $x=3$, and a local maximum near $(0.5,33)$. 2. **Recall the general form:** A quartic polynomial can be written as $$f(x) = a(x-r_1)^2(x-r_2)^2$$ if it has double roots at $r_1$ and $r_2$, which creates the W shape with touches at the x-axis. 3. **Apply the roots:** Given the roots at $x=-2$ and $x=3$, the function is $$f(x) = a(x+2)^2(x-3)^2$$ 4. **Determine the leading coefficient $a$ using the local maximum:** The local maximum near $(0.5,33)$ means $$f(0.5) = 33$$ Calculate: $$f(0.5) = a(0.5+2)^2(0.5-3)^2 = a(2.5)^2(-2.5)^2 = a(6.25)(6.25) = a(39.0625)$$ Set equal to 33: $$a \times 39.0625 = 33$$ 5. **Solve for $a$:** $$a = \frac{33}{39.0625} = 0.845$$ 6. **Final function:** $$f(x) = 0.845(x+2)^2(x-3)^2$$ This function matches the W-shaped quartic curve with the given intercepts and local maximum.