1. The problem statement, all variables and given/known data A farmer has 2400 feet of fencing and want to fence of a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area ? 2. Relevant equations We wish to maximize the area A of the rectangle. Let x and y be the width and length of the rectangle (in feet). Then we express A in terms of x and y as A = xy. We want to express A as a function of just one variable, so we eliminate y by expressing it in terms of x. To do this we use the given information that the total length of the fencing is 2400 ft. Therefore 2x + y = 2400 Hence y = 2400 − 2x and the area is A= x (2400 – 2x) = 2400 x − 2x2 Note that x ≥ 0 and x ≤ 1200 (otherwise A < 0). So the function that we wish to maximize is A (x) = 2400 x − 2x2, 0 ≤ x ≤ 1200. 3. The attempt at a solution A′(x) = 2400 − 4x, so to find the critical numbers we solve the equation 2400 − 4x = 0 which gives x = 600. The maximum of A must occur either at this critical number or at an end point of the interval. Since A(0) = 0, A(600) = 7,20,000 and A(1200) = 0, thus the maximum value is A (600) = 720,000. When x = 600, y = 2400 − 1200 = 1200. but my teacher insists me to solve the problem using second derivative test so,A''(x) = − 4 after this what should we do? since the second derivative of x is negative so it is a local maximum similarly the second derivative of y is also negative so how to find the x and y values?