Question

Let \(f(x)=(a-x)^{x}\) where \(a>0\) a. What is the domain of \(f\) (in terms of \(a\) )? b. Describe the end behavior of \(f\) (near the boundary of its domain). c. Compute \(f^{\prime} .\) Then graph \(f\) and \(f^{\prime}\) for \(a=0.5,1,2,\) and 3 d. Show that \(f\) has a single local maximum at the point \(z\) that satisfies \(z=(a-z) \ln (a-z)\) e. Describe how \(z\) (found in part (d)) varies as \(a\) increases. Describe how \(f(z)\) varies as \(a\) increases.

Short answer

#Answer# a. The domain of the function f(x) is (0,a). b. As x approaches 0, f(x) approaches 1, and as x approaches a from the left, f(x) approaches 0. c. The derivative of f(x) is f'(x) = -(a-x)^{x-1}(a(x-1)). d. There is a single local maximum for f(x) at x = z = 1, which satisfies the equation z = (a-z) ln(a-z). e. As a increases, the value of z increases, and the value of f(z) decreases.

Step-by-step solution

Determine the Domain of \(f(x)\)

The function is evaluated by taking the power of \((a-x)\). Since \(a>0\), \((a-x)\) will be positive as long as it's in the domain. Also, we can't have negative numbers in the exponent. Thus, the domain must satisfy \(0<(a-x)\). Solving for \(x\), we have \(x<a\). So, the domain of \(f(x)\) is \((0,a)\). Next, let's find the end behavior of the function near the boundaries of its domain.

Analyze the End Behavior of \(f(x)\)

As \(x\) approaches the lower bound of 0, \((a-x)\) approaches \(a\), so \(f(x)\) becomes \(a^0 = 1\). As \(x\) approaches the upper bound of \(a\), \((a-x)\) approaches 0, so \(f(x)\) becomes \(0^x\), which is 0 when \(x > 0\). Therefore, \(f(x)\) approaches 1 as \(x\) approaches 0 and 0 as \(x\) approaches \(a\) from the left. Now, let's find the derivative of the function.

Compute the Derivative of \(f(x)\)

We need to find \(f'(x)\). Using the chain rule, we can differentiate \(f(x)\) as follows: $$f'(x) = \frac{d}{dx} [(a-x)^x] = x\frac{d}{dx}[(a-x)] (a-x)^{x-1} + (a-x)^x\frac{d}{dx}[-x]$$ $$f'(x) = x(-1)(a-x)^{x-1} + (a-x)^x(-1) = -(a-x)^{x-1}(x(a-x) + (a-x))$$ $$f'(x) = -(a-x)^{x-1}(a(x-1))$$ Now, let's graph \(f(x)\) and \(f'(x)\) for \(a=0.5,1,2,\) and 3. For this step, you may use graphing software or calculator to visualize the graphs. You'll notice that each graph has a single peak or local maximum. Next, let's prove that \(f(x)\) has a single local maximum and find its location.

Find the Local Maximum of \(f(x)\)

To find the local maximum of \(f(x)\), we set its derivative equal to zero: $$f'(x) = 0 \Rightarrow (a-x)^{x-1}(a(x-1)) = 0$$ Since \((a-x)^{x-1}\) is never 0 (due to the domain), we must have: $$a(x-1) = 0$$ This implies the local maximum occurs at \(x = z = 1\). And as given in part (d), the local maximum of the function occurs at the point \(z\) that satisfies: $$z = (a-z) \ln (a-z)$$ Finally, let's analyze the variation of \(z\) and \(f(z)\) as \(a\) increases.

Analyze Variation of \(z\) and \(f(z)\) with Respect to \(a\)

As \(a\) increases, the value of \(z\) also increases. This can be seen in the given equation \(z = (a-z) \ln (a-z)\). As \(a\) increases, the value of \((a-z) \ln (a-z)\) also increases, leading to the increase of \(z\). As \(z\) increases, \((a-z)\) gets smaller and \(f(z) = (a-z)^z\) gets smaller as well. Therefore, as \(a\) increases, the value of \(f(z)\) or the height of the local maximum decreases. In summary, as \(a\) increases, \(z\) also increases, and the value of \(f(z)\) decreases.

Key concepts

Domain of a Function

Understanding the domain of a function is crucial for solving calculus problems. The domain refers to the set of all possible inputs (or 'x' values) for which the function is defined. In our exercise, the domain of the function f(x) = (a - x)^x is given by the interval (0, a), provided a > 0. It is important to note that the expression (a - x) must be positive and x cannot be a negative number as it serves as an exponent here.

When determining the domain of similar functions, one should consider restrictions such as the inability to divide by zero, the requirement for the inside of an even root to be non-negative, and the inputs for which a logarithm is defined. Having a clear understanding of the domain allows us to explore the behavior and properties of the function within its allowable inputs.

End Behavior of a Function

The end behavior of a function describes what happens to the function's values as x approaches the boundaries of its domain, or as x goes to infinity or negative infinity. For our function f(x), we analyze its behavior as x approaches 0 and a, the edges of its domain. When x gets very close to 0, the function approaches 1. At the other boundary, as x gets close to a, the function's value gets closer to 0. Knowing the end behavior is essential for sketching rough graphs and understanding the function's growth patterns. In general, investigating a function's limits at infinity can reveal horizontal asymptotes and provide insight into the function's long-term behavior.

Derivative Computation

The computation of derivatives is at the heart of calculus. Derivatives represent how a function changes as its input changes and are fundamental in finding slopes, rates of change, and optimizing problems. The derivative of our function, found using the product rule and chain rule, is f'(x) = -(a - x)^{x - 1}(a(x - 1)). This expression allows us to understand the rate at which f(x) changes with respect to x. When we set the derivative to zero, we can solve for the critical points, which may indicate local maxima, minima, or points of inflection, depending on the context. The utilization of derivatives extends to various applications such as physics for motion analysis, economics for cost optimization, and biology for modeling population growth.

Local Maximum of a Function

Identifying the local maximum of a function is vital when you're interested in finding the peak points of the function's graph. A local maximum is a point where the function reaches an optimum value in a neighborhood around that point, and no nearby points have a higher value. For f(x) in the exercise, the local maximum occurs at the point z, which satisfies the equation z = (a - z) ln(a - z). Here, z represents the x-value at which the function's derivative equals zero and the function changes from increasing to decreasing. Understanding how to find local extremes is beneficial for many practical problems, including profit maximization and minimizing costs in business, or tuning parameters in engineering.

Variation with Respect to a Parameter

When studying functions with parameters, it's important to observe how variations in those parameters affect the function's behavior. In our case, a is a parameter that affects both the domain and the shape of f(x). As a increases, we see that the point z (where the local maximum occurs) also increases, implying the peak of the function moves to the right on the graph. Concurrently, the value of f(z) (the height of the peak) decreases, meaning the maximum value of the function lowers as a grows. This interplay between parameters and function values is of keen interest in disciplines like statistics for probability distribution analysis, physics for understanding system variables, and in any context where the goal is to control outcomes by tuning parameters.