To confirm this is a maximum, note that $ C'(t) > 0 $ for $ 0 < t < 2 $ and $ C'(t) < 0 $ for $ t > 2 $, so the function increases then decreases. Thus, the maximum occurs at $ t = 2 $.

["Title: Understanding the Maximum of Function ( C(t) ): A Critical Analysis of Derivative Behavior", "Understanding when a function reaches its maximum value hinges largely on analyzing its derivative. In particular, the sign changes of ( C'(t) ) reveal whether a function is increasing or decreasing—and a precise knowledge of these transitions helps pinpoint exact points of extremum. This article explores a scenario where ( C'(t) > 0 ) for ( 0 < t < 2 ) and ( C'(t) < 0 ) for ( t > 2 ), demonstrating how this pattern confirms that the maximum of ( C(t) ) occurs at ( t = 2 ).", "### The Role of Derivatives in Determining Maximums", "In calculus, a function’s derivative ( C'(t) ) represents its instantaneous rate of change. When ( C'(t) > 0 ), the function is increasing; conversely, when ( C'(t) < 0 ), the function is decreasing. Therefore, a change in derivative sign—from positive to negative—at a point indicates a local maximum, assuming continuity and differentiability near that point.", "In the example discussed, ( C'(t) > 0 ) on the open interval ( (0, 2) ) shows ( C(t) ) is rising during this time. Once ( t ) exceeds 2, ( C'(t) < 0 ) signals a decrease. The crossover from positive to negative derivative at exactly ( t = 2 ) confirms definitively that ( C(t) ) reaches its highest value at this moment.", "### Why ( t = 2 ) Is the Absolute Maximum", "Since ( C(t) ) increases steadily before ( t = 2 ) and begins decreasing afterward, the function cannot rise further or fall to lower values after ( t = 2 ). The point ( t = 2 ) is thus a global maximum on the interval under consideration, provided no larger values occur elsewhere.", "This method is powerful in optimization contexts such as maximizing profit, area, or other real-world quantities modeled by differentiable functions. Understanding how derivative signs change empowers precise identification of optimal points without requiring explicit evaluation of every function value.", "### Conclusion", "Confirming maximums via derivative sign changes is both elegant and effective. When ( C'(t) ) transitions from positive to negative precisely at ( t = 2 ), it leaves no doubt: ( C(t) ) attains its maximum at that precise moment. This analytical approach is foundational in calculus and critical applications across science, engineering, and economics.", "If you're analyzing function maxima, always examine the derivative’s sign pattern—its transverse shifts from positive to negative often mark the pinnacle of quantified behavior.", "---", "Keywords:\nmaximum of a function, derivative analysis, ( C'(t) > 0 ), ( C'(t) < 0 ), critical points, calculus optimization, local and global maxima, increasing then decreasing function, function behavior.", "Meta Description:\nDiscover how analyzing the sign changes of ( C'(t) ) reveals maximum values in differentiable functions. Learn why ( t = 2 ) is the maximum when ( C'(t) > 0 ) for ( 0 < t < 2 ) and ( C'(t) < 0 ) for ( t > 2 )."]