A quadratic production model
A student models the height of a small display panel by h(x)=−x²+6x+1 for a horizontal coordinate x measured in meters. The model is intended only for 0≤x≤4. A classmate finds a stationary point but does not compare it with the endpoints, while another assumes that the constant term must be the maximum height. The teacher asks for a complete mathematical explanation using the stated interval rather than a sketch alone. There is no additional physical condition restricting the panel within that interval. All lengths are measured on the same scale, and no rounding is needed for the exact answer. The task is a supplied practice model rather than a statement about a real construction specification.
1. Find the largest height in the model.
Read the worked answer
h′(x)=−2x+6 vanishes at x=3. Compare h(0)=1, h(3)=10 and h(4)=9. The maximum is 10 meters within the stated interval.
2. Explain why checking the interval matters.
Read the worked answer
A stationary point outside the allowed interval would not be a permissible candidate. Endpoint values also matter for a closed interval, even if the unrestricted curve has a familiar vertex.