Maximize Area of Rectangle w/ 2400ft Fence

[smart_worker]

Homework Statement

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 ?

Homework 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.

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?

 

[DrClaude]

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.

Note that $A'(x)$ has a single root, therefore it is either the maximum you are looking for, or there is no solution (except for $x \rightarrow \pm\infty$). You therefore only need to show that $x=600$ is a maximum. Your teacher wants you to do that using $A''(x)$. You obtain the final answer the same way you already have.