Check the second derivative or evaluate \(C(t)\) at critical points and endpoints to confirm maximum:

["Confirming Maximum Values: A Step-by-Step Guide Using the Second Derivative and Critical Points", "When analyzing functions in calculus—such as in optimization, physics, or economics—determining whether a point yields a maximum value is essential. One powerful method combines checking the second derivative with evaluating critical points and endpoints. This article explains how to confirm a maximum using these tools and why they work together.", "---", "### Why Confirm a Maximum?", "Identifying maximum values helps solve real-world problems like maximizing profit, optimizing area, or analyzing peak values in physical systems. While first derivatives tell us where extrema occur, second derivatives clarify which of those points are maximum.", "---", "### Step 1: Find Critical Points", "A critical point is a value ( t ) in the domain where the first derivative ( C'(t) = 0 ) or where ( C'(t) ) doesn’t exist. Often, these points are candidates for local maxima, minima, or saddle points.", "Process:\n1. Differentiate ( C(t) ) to find ( C'(t) ).\n2. Solve ( C'(t) = 0 ) to find critical points.\n3. Include points where ( C'(t) ) is undefined (if applicable).", "---", "### Step 2: Analyze the Second Derivative", "The second derivative test helps categorize critical points:", "- If ( C''(t) < 0 ) at a critical point, the function is concave down → the point is a local maximum.\n- If ( C''(t) > 0 ), the point is a local minimum.\n- If ( C''(t) = 0 ), the test is inconclusive—further analysis is needed.", "This works because the sign of ( C''(t) ) reflects the curvature of ( C(t) ): negative curvature indicates a peak.", "---", "### Step 3: Evaluate Endpoints (for Closed Intervals)", "If the domain is a closed interval ([a, b]), examine the function at the endpoints ( t = a ) and ( t = b ). Sometimes maxima occur at boundaries rather than critical points, especially when the function isn’t defined beyond the interval.", "- Compute ( C(a) ) and ( C(b) ).\n- Compare these values with ( C(t) ) at critical points.\n- The largest value identifies the global maximum within ([a, b]).", "---", "### Practical Example", "Consider a function modeling revenue over time: ( C(t) = -2t^3 + 9t^2 + 12t )", "1. Finding critical points:\n ( C'(t) = -6t^2 + 18t + 12 )\n Set ( C'(t) = 0 ):\n [\n -6t^2 + 18t + 12 = 0 \quad \Rightarrow \quad t^2 - 3t - 2 = 0\n ]\n Using the quadratic formula, ( t \approx 3.646 ) and ( t \approx -0.646 ).\n Only ( t \approx 3.646 ) lies in the domain (say, ( t \in [0, 5] )).", "2. Second derivative test:\n Compute ( C''(t) = -12t + 18 ).\n At ( t \approx 3.646 ):\n [\n C''(3.646) \approx -12(3.646) + 18 \approx -29.35 < 0\n ]\n Since ( C''(t) < 0 ), this critical point is a local maximum.", "3. Evaluate endpoints:\n - ( C(0) = 0 )\n - ( C(5) = -2(125) + 9(25) + 60 = -250 + 225 + 60 = 35 )\n - ( C(3.646) \approx -2(48.66) + 9(13.30) + 44.1 \approx -97.32 + 119.7 + 44.1 \approx 66.48 )", "Since ( C(3.646) \approx 66.48 ) is greater than values at ( 0 ) and ( 5 ), it is the global maximum on ([0, 5]).", "---", "### Summary", "- Critical points reveal potential extrema via derivative analysis.\n- The second derivative confirms whether a critical point is a local maximum.\n- Endpoints must be checked in closed intervals to locate global maxima.", "Combining these methods ensures precise identification of maximum values—key for effective mathematical modeling and decision-making.", "---", "#### Final Tips\n- When analytic derivatives are complex, numerical methods or graphing aids should supplement the test.\n- Always verify continuity and differentiability on closed intervals.\n- In applied fields, understanding the physical meaning of the function enhances interpretation of maxima.", "---", "Keywords: second derivative test, critical points, maximum confirmation, calculus optimization, evaluating endpoints, function analysis, real-world applications.", "By mastering these techniques, you equip yourself to reliably determine maximum values—a cornerstone of calculus in both theory and practice."]