Solution: The minimum occurs at $ x = -\frac{b}{2a} = -\frac{-20}{2(5)} = 2 $. Verify $ W(2) = 5(4) - 20(2) + 45 = 20 - 40 + 45 = 25 $. The smallest $ x $ is $ \boxed{2} $.Question: A hydrologist models groundwater flow with the function $ f(x) = (\sec x + \cos x)^2 + (\csc x - \sin x)^2 $ for $ 0 < x < \frac{\pi}{2} $. Find the minimum value of $ f(x) $.

["Solution: Find the Minimum of $ f(x) = (\sec x + \cos x)^2 + (\csc x - \sin x)^2 $ on $ \left(0, \frac{\pi}{2}\right) $", "Optimizing functions in physical modeling—such as groundwater flow dynamics—requires identifying critical points where change in behavior occurs. In this case, we analyze the trigonometric function\n[\nf(x) = (\sec x + \cos x)^2 + (\csc x - \sin x)^2\n]\nfor $ x \in \left(0, \frac{\pi}{2}\right) $. Our goal is to find its minimum value.", "### Step 1: Use the given insight about the vertex", "From classical calculus, the function $ f(x) $ achieves its minimum when the expression simplifies near $ x = -\frac{b}{2a} $—a principle seen in quadratic models. Here, although $ f(x) $ is not quadratic, the structure of symmetry and known behavior at critical points suggests a minimum at a value analogous to $ x = 2 $ in the earlier example. However, our domain is $ \left(0, \frac{\pi}{2}\right) \approx (0, 1.57) $. So we evaluate if $ x = \frac{\pi}{4} $ or $ x = 1 $ (nearest simple value) yields the minimum—leading us to test the given candidate.", "But let’s verify the minimum value by analyzing the function algebraically and using calculus to confirm.", "### Step 2: Simplify the function", "We rewrite using identities:", "- $ \sec x = \frac{1}{\cos x} $\n- $ \csc x = \frac{1}{\sin x} $", "So:\n[\nf(x) = \left(\frac{1}{\cos x} + \cos x\right)^2 + \left(\frac{1}{\sin x} - \sin x\right)^2\n]", "Expand both terms:", "First term:\n[\n\left(\frac{1}{\cos x} + \cos x\right)^2 = \frac{1}{\cos^2 x} + 2 + \cos^2 x\n]", "Second term:\n[\n\left(\frac{1}{\sin x} - \sin x\right)^2 = \frac{1}{\sin^2 x} - 2 + \sin^2 x\n]", "Add:\n[\nf(x) = \left(\frac{1}{\cos^2 x} + \cos^2 x + 2\right) + \left(\frac{1}{\sin^2 x} + \sin^2 x - 2\right)\n]\n[\nf(x) = \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} + \cos^2 x + \sin^2 x + (2 - 2)\n]", "Since $ \cos^2 x + \sin^2 x = 1 $, we get:\n[\nf(x) = \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} + 1\n]", "Use identities: $ \frac{1}{\cos^2 x} = \sec^2 x = 1 + \ an^2 x $, $ \frac{1}{\sin^2 x} = \csc^2 x = 1 + \cot^2 x $, but better to combine:", "[\nf(x) = \sec^2 x + \csc^2 x + 1\n]", "Now, recall:\n[\n\sec^2 x + \csc^2 x = \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} = \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}\n]", "So:\n[\nf(x) = \frac{1}{\sin^2 x \cos^2 x} + 1\n]", "Use identity: $ \sin(2x) = 2\sin x \cos x \Rightarrow \sin x \cos x = \frac{1}{2} \sin 2x $, so:\n[\n\sin^2 x \cos^2 x = \left(\frac{1}{2} \sin 2x\right)^2 = \frac{1}{4} \sin^2 2x\n]", "Thus:\n[\nf(x) = \frac{1}{\frac{1}{4} \sin^2 2x} + 1 = \frac{4}{\sin^2 2x} + 1\n]", "### Step 3: Minimize $ f(x) = \frac{4}{\sin^2 2x} + 1 $", "Since $ x \in \left(0, \frac{\pi}{2}\right) $, then $ 2x \in (0, \pi) $, and $ \sin 2x > 0 $. The function $ \sin^2 2x $ reaches maximum 1 when $ 2x = \frac{\pi}{2} \Rightarrow x = \frac{\pi}{4} $.", "At $ x = \frac{\pi}{4} $, $ \sin^2 2x = \sin^2 \left(\frac{\pi}{2}\right) = 1 $, so:\n[\nf\left(\frac{\pi}{4}\right) = \frac{4}{1} + 1 = 5\n]", "Check second derivative or behavior: as $ x \ o 0^+ $ or $ x \ o \frac{\pi}{2}^- $, $ \sin 2x \ o 0 $, so $ f(x) \ o \infty $. Thus, the minimum occurs at $ x = \frac{\pi}{4} $.", "### Final Answer", "The minimum value of $ f(x) $ is $ \boxed{5} $.\nThis value reflects optimal efficiency in energy dispersion across the modeled flow domain—consistent with natural equilibrium states in hydrological systems."]