![]() ![]() The answer to all of your questions is: yes! If the second derivative is a negative constant, then the function is concave down everywhere, and so youâre guaranteed that the point x=c you found where f'(c) = 0 is a maximum. And agreed about getting the problem set-up right as the vast majority of the work here. Weâre glad to know you liked our explanation and approach. Hereâs a key thing to know about how to solve Optimization problems: youâll almost always have to use detailed information given in the problem to rewrite the equation you developed in Step 2 to be in terms of one single variable.Ībove, for instance, our equation for $A_\text \quad \cmark Optimization Problems & Complete Solutions.This minimum must occur at a critical point of S. Since S is a continuous function that approaches infinity at the ends, it must have an absolute minimum at some x\in (0,\infty ). Step 6: Note that as x\to 0^+, S(x)\to \infty. However, in the next step, we discover why this function must have an absolute minimum over the interval (0,\infty ). Note that, unlike the previous examples, we cannot reduce our problem to looking for an absolute maximum or absolute minimum over a closed, bounded interval. We conclude that the domain is the open, unbounded interval (0,\infty ). Similarly, as x becomes small, the height of the box becomes correspondingly large. Note that as x becomes large, the height of the box y becomes correspondingly small so that x^2 y=216. On the other hand, x is allowed to have any positive value. Step 5: Since we are requiring that x^2 y=216, we cannot have x=0. Since V(x)=0 at the endpoints and V(x)>0 for 0 ![]() 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 in. Step 5: To determine the domain of consideration, letâs examine Figure 3. Now letâs 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 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. Write any equations relating the independent variables in the formula from step 3.This formula may involve more than one variable. Write a formula for the quantity to be maximized or minimized in terms of the variables.Determine which quantity is to be maximized or minimized, and for what range of values of the other variables (if this can be determined at this time).If applicable, draw a figure and label all variables. Problem-Solving Strategy: Solving Optimization Problems Differentiating the function A(x), we obtain Since the area is positive for all x in the open interval (0,50), the maximum must occur at a critical point. ![]() These extreme values occur either at endpoints or critical points. Maximize A(x)=100x-2x^2 over the interval Īs mentioned earlier, since A is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum. Therefore, we consider the following problem: If the maximum value occurs at an interior point, then we have found the value x in the open interval (0,50) that maximizes the area of the garden. Therefore, letâs consider the function A(x)=100x-2x^2 over the closed interval. However, we do know that a continuous function has an absolute maximum (and absolute minimum) over a closed interval. ![]() We do not know that a function necessarily has a maximum value over an open interval. Therefore, we are trying to determine the maximum value of A(x) for x over the open interval (0,50). To construct a rectangular garden, we certainly need the lengths of both sides to be positive. A(x)=x \cdot (100-2x)=100x-2x^2Ä«efore trying to maximize the area function A(x)=100x-2x^2, we need to determine the domain under consideration. ![]()
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