Calculus problems3/23/2023 ![]() ![]() Since V ( x ) = 0 V ( x ) = 0 at the endpoints and V ( x ) > 0 V ( x ) > 0 for 0 < x < 12, 0 < x < 12, the maximum must occur at a critical point. Step 6: Since V ( x ) V ( x ) is a continuous function over the closed, bounded interval, , V V must have an absolute maximum (and an absolute minimum). Therefore, we consider V V over the closed interval and check whether the absolute maximum occurs at an interior point. Since V V is a continuous function over the closed interval, , we know V V will have an absolute maximum over the closed interval. Therefore, we are trying to determine whether there is a maximum volume of the box for x x over the open interval ( 0, 12 ). otherwise, one of the flaps would be completely cut off. Furthermore, the side length of the square cannot be greater than or equal to half the length of the shorter side, 24 24 in. Step 5: To determine the domain of consideration, let’s examine Figure 4.64. Now let’s apply this strategy to maximize the volume of an open-top box given a constraint on the amount of material to be used. This step typically involves looking for critical points and evaluating a function at endpoints. Locate the maximum or minimum value of the function from step 4.Identify the domain of consideration for the function in step 4 4 based on the physical problem to be solved.Use these equations to write the quantity to be maximized or minimized as a function of one variable. ![]()
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