Maximize F(x) = X^a(9-x)^b On [0,9]: A Step-by-Step Guide

by Andrew McMorgan 58 views

Hey guys! Today, we're diving deep into a super interesting problem: finding the maximum value of a polynomial function within a specific interval. Specifically, we'll be tackling the function f(x) = xa(9-x)b on the interval [0, 9], where a and b are positive constants. This might sound a bit intimidating at first, but don't worry, we'll break it down step-by-step, making sure everyone can follow along. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we fully understand what the problem is asking. We're given a function, f(x) = xa(9-x)b, which is a polynomial function since a and b are positive constants. This means that the exponents are positive numbers, and there are no fractional or negative exponents involved. The function itself is a product of two terms: x^a and (9-x)^b. Each of these terms will influence the behavior of the function, and understanding how they interact is key to finding the maximum value.

The interval [0, 9] is crucial because it restricts the domain of x. We're only interested in the values of x that fall between 0 and 9, inclusive. This means we need to consider the endpoints (0 and 9) as well as any points within the interval where the function might reach its maximum. Think of it like searching for the highest point on a mountain range within a specific region – we need to look at the peaks within that region and also the edges to make sure we haven't missed anything.

The Main Objective: Our primary goal is to determine the largest possible value that f(x) can attain within the interval [0, 9]. This means finding the x-value that, when plugged into the function, yields the highest f(x) value. This value is often referred to as the global maximum within the given interval. To achieve this, we'll leverage the power of calculus, specifically derivatives, which will help us pinpoint critical points where the function's slope changes, potentially indicating a maximum or minimum.

Finding Critical Points Using Derivatives

The heart of solving this problem lies in using calculus, specifically the concept of derivatives. Derivatives are powerful tools that allow us to find the rate of change of a function. In simpler terms, the derivative tells us how the function's output changes as the input changes. At a maximum or minimum point of a function, the rate of change is zero (think of the peak of a hill – the slope is momentarily flat). These points where the derivative is zero (or undefined) are called critical points, and they are our prime suspects for where the maximum value might occur.

Step 1: Finding the Derivative

To find the critical points, we first need to calculate the derivative of f(x). Since our function is a product of two terms, we'll need to use the product rule. The product rule states that the derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x). Let's apply this to our function:

  • Let u(x) = x^a, then u'(x) = ax^(a-1)
  • Let v(x) = (9-x)^b, then v'(x) = -b(9-x)^(b-1)

Now, applying the product rule, we get:

f'(x) = ax(a-1)(9-x)b - bxa(9-x)(b-1)

This derivative looks a bit complex, but we're not done yet. We need to simplify it to make it easier to work with.

Step 2: Simplifying the Derivative

The key to simplifying the derivative is to factor out common terms. Notice that both terms in f'(x) have x^(a-1) and (9-x)^(b-1) as factors. Factoring these out, we get:

f'(x) = x(a-1)(9-x)(b-1) [a(9-x) - bx]

This simplified form is much easier to handle. Now, we can find the critical points by setting f'(x) equal to zero.

Step 3: Finding Critical Points

To find the critical points, we set f'(x) = 0:

x(a-1)(9-x)(b-1) [a(9-x) - bx] = 0

For this equation to be true, at least one of the factors must be zero. This gives us three possibilities:

  1. x^(a-1) = 0 => x = 0 (assuming a > 1)
  2. (9-x)^(b-1) = 0 => x = 9 (assuming b > 1)
  3. a(9-x) - bx = 0 => 9a - ax - bx = 0 => x = 9a / (a + b)

So, we have three potential critical points: 0, 9, and 9a / (a + b). These are the points where the function's slope might change, and therefore where we might find a maximum or minimum.

Evaluating the Function at Critical Points and Endpoints

Now that we have our critical points, we need to evaluate the function f(x) at these points, as well as at the endpoints of the interval [0, 9]. This will allow us to compare the function values and identify the absolute maximum within the interval.

Step 1: Evaluating at Endpoints

Let's start by evaluating f(x) at the endpoints of the interval:

  • f(0) = 0a(9-0)b = 0
  • f(9) = 9a(9-9)b = 9a(0)b = 0

So, at both endpoints, the function value is 0. This makes sense, as the x^a term will be zero when x is 0, and the (9-x)^b term will be zero when x is 9. Now, let's move on to the critical point in the middle.

Step 2: Evaluating at the Critical Point

We found a critical point at x = 9a / (a + b). This is the most interesting point, as it's likely where the maximum value occurs. Let's plug this value into our function:

f(9a / (a + b)) = (9a / (a + b))^a (9 - 9a / (a + b))^b

This looks complicated, but we can simplify it. Let's focus on the second term:

9 - 9a / (a + b) = (9(a + b) - 9a) / (a + b) = 9b / (a + b)

Now, we can substitute this back into our expression for f(9a / (a + b)):

f(9a / (a + b)) = (9a / (a + b))^a (9b / (a + b))^b

This can be further simplified by combining the terms with the same denominator:

f(9a / (a + b)) = (9^a a^a / (a + b)^a) (9^b b^b / (a + b)^b) = 9^(a+b) a^a b^b / (a + b)^(a+b)

This is the value of the function at our critical point. Now, we need to compare this value with the values at the endpoints (which were both 0) to determine the maximum.

Step 3: Comparing Values and Determining the Maximum

We have three values to compare:

  • f(0) = 0
  • f(9) = 0
  • f(9a / (a + b)) = 9^(a+b) a^a b^b / (a + b)^(a+b)

Since a and b are positive constants, the value of f(9a / (a + b)) will always be greater than 0. Therefore, the maximum value of the function on the interval [0, 9] is:

f_max = 9^(a+b) a^a b^b / (a + b)^(a+b)

This maximum occurs at x = 9a / (a + b).

Conclusion

Alright, guys! We've successfully navigated the problem of finding the maximum value of f(x) = xa(9-x)b on the interval [0, 9]. We used the power of derivatives to identify critical points, evaluated the function at these points and the endpoints, and ultimately determined the maximum value and where it occurs. This process highlights the beauty and utility of calculus in solving optimization problems. Remember, breaking down complex problems into smaller, manageable steps is the key to success. And who knows, maybe this knowledge will come in handy when you're optimizing your next project or even planning a road trip!