7.3: Checking Endpoints
- Page ID
- 121120
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- Recognize the distinction between local and global extrema.
- Find the global minimum or maximum in a given word problem.
In some cases, the optimal value of a function does not occur at any of its local maxima, but rather at one of the endpoints of an interval. Here we consider such an example.
The area of a rectangle with side lengths \(x\) and \(y\) is \(A=x y\). Suppose that the variable \(x\) is only allowed to take on values in the range \(0.5 \leq x \leq 4\). Find the dimensions of the rectangle having largest perimeter whose fixed area is \(A=1\).
Solution
The perimeter of a rectangle whose sides are length \(x, y\) is
\[P=x+y+x+y=2 x+2 y . \nonumber \]
We are to maximize this quantity subject to the area of the rectangle being fixed, \(A=x y=1\). This is the constraint. We use it to solve for and to eliminate \(y\) from \(P\)
\[y=\frac{1}{x}, \quad \Rightarrow \quad P(x)=2 x+\frac{2}{x} . \nonumber \]
We look for \(x\) that maximizes \(P(x)\). Computing the derivatives,
\[P^{\prime}(x)=2\left(1-\frac{1}{x^{2}}\right), \quad P^{\prime \prime}(x)=\frac{4}{x^{3}}>0 \nonumber \]
Setting \(P^{\prime}(x)=0\), we find critical points satisfying \(x^{2}=1\) or \(x=\pm 1\). We reject the negative root as irrelevant. We have found that \(P^{\prime \prime}(x)>0\) for all \(x>0\), so the critical point is a local minimum! This is clearly not the maximum we were looking for. This example reinforces the importance of diagnostic tests for the type of critical point.
Next, checking the endpoints of the interval, we evaluate \(P(4)=8.5\) and \(P(0.5)=5\). The largest perimeter for the rectangle thus occurs when \(x=4\), at the right endpoint of the domain, as shown in Figure 7.5.
- How do you calculate the perimeter of a rectangle?
- Why can a negative root of \(P^{\prime}(x)=0\) in Example \(7.5\) be rejected as irrelevant?
In Appendix G.4, we provide further examples of optimization in the context of geometric solids.
- Use Figure \(7.5\) to estimate the side length \(x\) when \(P(x)=6\).
- Verify your estimate algebraically.