1. We are given the function $p(u) = u^3 - 3u^2 + 4u$ with $u$ in meters. The problem involves understanding this cubic function and its interaction with the factor $(u - 2)$, which may represent a factorization or a root related to the raw material in the shoe. 2. First, factor $p(u)$ if possible. Factor out the common term $u$: $$p(u) = u(u^2 - 3u + 4)$$ 3. Next, analyze the quadratic $u^2 - 3u + 4$: Calculate its discriminant: $$\Delta = (-3)^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7$$ Because $\Delta < 0$, the quadratic has no real roots, so $p(u)$ has only one real root from the factor $u = 0$. 4. The factor $(u - 2)$ given might be a proposed factor of $p(u)$ or related to raw material input. To check if $(u - 2)$ is a factor, evaluate $p(2)$: $$p(2) = 2^3 - 3 \cdot 2^2 + 4 \cdot 2 = 8 - 12 + 8 = 4$$ Since $p(2) \ne 0$, $(u - 2)$ is not a factor. 5. To summarize the behavior: - The curve crosses the $u$-axis at $u=0$. - The cubic has no other real roots. - The raw material $(u - 2)$ does not represent a root or factor for $p(u)$. 6. The local extrema can be found by taking the derivative: $$p'(u) = 3u^2 - 6u + 4$$ Set $p'(u) = 0$: $$3u^2 - 6u + 4 = 0$$ Calculate discriminant: $$\Delta = (-6)^2 - 4 \cdot 3 \cdot 4 = 36 - 48 = -12$$ Since $\Delta < 0$, no critical points with real roots exist, so $p(u)$ has no local maximum or minimum. The function is strictly increasing or decreasing throughout its domain. Final answer: - $p(u)$ has one real root at $u=0$. - $(u - 2)$ is not a factor. - No local extrema exist. - The graph’s shape confirms this behavior.