91Ó°ÊÓ

In optimization problems, the objective function is a vital element that determines what is being optimized. Here, the objective function is the energy expended by the bird per day, denoted as \( E \). It depends on the time spent foraging for food, expressed as \( F \) hours.
The function is given by \( E = 0.25F + \frac{1.7}{F^2} \). This equation suggests that energy expenditure increases linearly with foraging time, \( F \), and also includes an inverse cubic term \( \frac{1.7}{F^2} \) attributed to territorial defense. Our task in this scenario is to find the optimal \( F \) that minimizes the energy \( E \).
By analyzing and altering the objective function, we can better understand and refine the conditions under which the bird expends the least energy.

Calculating the derivative of the objective function is a crucial step in optimization. It helps to determine the rate of change of energy expenditure with changes in foraging time. To optimize \( E \), we need to find its derivative with respect to \( F \).
The original function \( E = 0.25F + \frac{1.7}{F^2} \) is differentiated to produce:
\[\frac{dE}{dF} = 0.25 - \frac{3.4}{F^3}\]
This derivative indicates the slope of the energy function at any given \( F \). A slope of zero at some point \( F \) suggests a potential minimum or maximum, leading us to the next step in our analysis.

Critical points in calculus are values of \( F \) where the derivative \( \frac{dE}{dF} = 0 \). To find these points, we set the derivative computed earlier to zero:
\[0.25 - \frac{3.4}{F^3} = 0\]
Solving gives us \( F^3 = \frac{3.4}{0.25} \), providing the critical point \( F = \sqrt[3]{13.6} \approx 2.38 \).
This critical point is the candidate for where the minimum energy expenditure occurs. It is essential to confirm that this critical point indeed represents a minimum by further analysis.

To confirm whether the critical point found provides a minimum, we use the second derivative test. The second derivative \( \frac{d^2E}{dF^2} \) assesses the concavity of the function:
\[\frac{d^2E}{dF^2} = \frac{10.2}{F^4}\]
Since \( \frac{10.2}{F^4} \) is positive for all \( F > 0 \), the energy function is concave up at \( F = 2.38 \), indicating a local minimum. Thus, this test confirms that a foraging time of approximately 2.38 hours minimizes the bird's daily energy expenditure.
This concludes the optimization procedure, providing an analytical approach to finding optimal conditions efficiently.