Question: A hydrologist is minimizing the energy cost of pumping water through an aquifer modeled by the function $f(x) = 2x^2 - 8x + 11$. Find the minimum value of $f(x)$.
["SEO Article: Minimizing Energy Cost in Aquifer Pumping Using Optimization of $f(x) = 2x^2 - 8x + 11$", "Hydrologists and engineers seeking to minimize energy costs associated with pumping water through aquifers often turn to mathematical modeling. A key quadratic function, $f(x) = 2x^2 - 8x + 11$, represents the theoretical energy requirement for pumping at various flow rates or depths. This article explores how to find the minimum value of this function, enabling efficient and cost-effective water extraction from subsurface reservoirs.", "---", "### Finding the Minimum Energy Cost: A Quadratic Optimization Problem", "In hydrological engineering, minimizing pumping energy is essential for sustainable groundwater management. The energy cost function, modeled by a quadratic equation like $f(x) = 2x^2 - 8x + 11$, typically has a U-shaped curve with a single minimum — representing the most energy-efficient pumping level.", "Let’s analyze and solve for the minimum value of:", "$$\nf(x) = 2x^2 - 8x + 11\n$$", "---", "### Step 1: Identify the Type of Function", "The function $f(x)$ is a quadratic function in the standard form:", "$$\nf(x) = ax^2 + bx + c\n$$", "Here, $a = 2$, $b = -8$, and $c = 11$. Since $a > 0$, the parabola opens upwards, confirming the presence of a minimum at its vertex.", "---", "### Step 2: Locate the Vertex", "The x-coordinate of the vertex — the point where the minimum occurs — is given by:", "$$\nx = -\frac{b}{2a} = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2\n$$", "---", "### Step 3: Compute the Minimum Value", "Substitute $x = 2$ back into the original function to find the minimum energy cost:", "$$\nf(2) = 2(2)^2 - 8(2) + 11 = 2(4) - 16 + 11 = 8 - 16 + 11 = 3\n$$", "---", "### Conclusion: The Minimum Energy Cost is 3", "Thus, the hydrologist finds that the lowest energy cost occurs at a pumping depth or rate modeled by $x = 2$, and the minimum value of $f(x)$ is:", "$$\n\boxed{3}\n$$", "---", "### Why This Matters for Sustainable Groundwater Use", "By leveraging mathematical models like $f(x) = 2x^2 - 8x + 11$, professionals can pinpoint optimal operating points that reduce energy expenditures. This not only cuts operational costs but also supports environmentally responsible groundwater extraction, crucial for preserving aquifer health in regions facing water scarcity.", "---", "Keywords: hydrologist, aquifer pumping, energy optimization, quadratic function, minimum value, $f(x) = 2x^2 - 8x + 11$, groundwater modeling, cost minimization, mathematical optimization, sustainable water extraction", "---", "Meta Description:\nDiscover how hydrologists minimize energy costs in aquifer pumping using the quadratic function $f(x) = 2x^2 - 8x + 11$. Find the exact minimum value and its real-world implications for efficient groundwater management."]
