Maximize & Minimize: $5*(M+m)$ Of $5^{x^2-4}$

Hey guys! Let's dive into a fun little math problem. We're tasked with finding the value of the expression $5 imes (M + m)$. Here, M represents the largest value, and m represents the smallest value of the function $f(x) = 5^{x^2 - 4}$. But there's a catch: we're only looking at this function on the interval from 3 to 6, which is written as $[3; 6]$. Sounds interesting, right? Don't worry, it's not as scary as it might seem at first glance. We'll break it down step by step. Our mission is to uncover the secrets hidden within the function and its behavior within a specific range.

Understanding the Function and the Interval

First off, let's get familiar with our star player: the function $f(x) = 5^{x^2 - 4}$. This is an exponential function, and the key thing to notice is that the exponent itself, $x^2 - 4$, is a quadratic function. This means we're dealing with a function that will change its rate of growth as x changes. To solve this, we need to figure out how the exponent changes as x goes from 3 to 6. Knowing this will help us determine where the function hits its highest and lowest points within that interval.

The interval $[3; 6]$ is like our playground. It tells us that x can only take on values between 3 and 6, including 3 and 6 themselves. Our goal is to find the maximum (M) and minimum (m) values of the function within this specific range. Keep in mind that exponential functions can be tricky, and their behavior depends heavily on the exponent's behavior. Because $f(x)$ is an exponential function with a base greater than 1 (5), the larger the exponent, the larger the function's value. This gives us a huge hint, meaning we need to focus on maximizing and minimizing the exponent, $x^2-4$. Now, the fun part is about to begin! So, let's roll up our sleeves and start finding the solution! Let's explore the inner workings of this function, and with a bit of logical thinking, we'll conquer this math challenge together.

Finding the Maximum and Minimum Values

Alright, let's get down to business and find those maximum and minimum values! Since our function is $f(x) = 5^{x^2 - 4}$, and we know the base (5) is greater than 1, the function's behavior is directly linked to the exponent, $x^2 - 4$. To find the maximum and minimum of our function, we need to check the behavior of the exponent within our given interval $[3; 6]$.

The exponent is a quadratic function, $x^2 - 4$. Since we are dealing with a parabola opening upwards, it means the vertex is the minimum point. But in our interval, since the parabola is always increasing, the minimum value will be at the beginning of the interval, which is x = 3. The maximum value will be at the end of the interval, which is x = 6. So, let's start by plugging in the endpoints of our interval to find the corresponding values of x:

• When x = 3, the exponent is $3^2 - 4 = 9 - 4 = 5$. This means $f(3) = 5^5$.
• When x = 6, the exponent is $6^2 - 4 = 36 - 4 = 32$. Therefore, $f(6) = 5^{32}$.

We've got $f(3) = 5^5$ and $f(6) = 5^{32}$. Clearly, $5^{32}$ is the larger value, and $5^5$ is the smaller one. Therefore, the maximum value (M) of the function on the interval $[3; 6]$ is $5^{32}$, and the minimum value (m) is $5^5$. Pretty straightforward, huh?

Calculating the Final Expression

Now that we have M and m, the last step is to find the value of the expression $5 imes (M + m)$. We already know:

• $M = 5^{32}$
• $m = 5^5$

So, let's substitute these values into the expression:

$5 imes (M + m) = 5 imes (5^{32} + 5^5)$.

Now, we calculate the sum inside the parenthesis: $5^{32} + 5^5$. We will then multiply the result by 5. Since the numbers are quite large, we'll leave the answer as $5 imes (5^{32} + 5^5)$. So the answer is:

$5 imes (5^{32} + 5^5)$.

And there you have it! We've successfully found the maximum and minimum values of the function and calculated the final expression. It might look complex at first, but by breaking it down into smaller, manageable steps, it becomes much easier. Congratulations, guys, we've cracked the code! This journey through the world of exponential functions and intervals has been a blast, don't you think? We've seen how the behavior of the exponent dictates the function's overall behavior, and how important the interval is in determining the function's extremes.

Summary of Results

Here is a concise summary of what we've achieved:

• Function: $f(x) = 5^{x^2 - 4}$
• Interval: $[3; 6]$
• Maximum Value (M): $5^{32}$ (at x = 6)
• Minimum Value (m): $5^5$ (at x = 3)
• Final Expression: $5 imes (M + m) = 5 imes (5^{32} + 5^5)$

We started with a seemingly complex function, understood its components, especially the role of the exponent, and systematically examined its behavior within the given interval. We found the maximum and minimum values by analyzing the exponent and then plugged those values into the final expression. This is a great example of how understanding the basic principles of functions and intervals can help you solve even the most challenging problems. Remember that the key is always to break down the problem into simpler steps and to pay close attention to the details. Well done, everyone!

Conclusion and Further Exploration

So, to wrap things up, we've found the value of the expression $5 imes (M + m)$, where M is the maximum and m is the minimum value of the function within the interval $[3; 6]$. We've seen how the interplay of the function's base and the exponent determines its behavior. The interval served as a crucial constraint, guiding us to the specific values we needed. This kind of problem highlights the importance of understanding not just the formulas, but also the underlying concepts of functions, exponents, and intervals.

For further exploration, you could try these exercises:

• Change the Interval: What would happen if the interval was different? Try solving the problem with a different interval, such as $[-1; 2]$ or $[0; 4]$. How do the maximum and minimum values change?
• Change the Function: What happens if we change the function to something like $f(x) = 2^{x^2 - 4}$ or $f(x) = 5^{x^3 - 4x}$? How does the change in the function affect the process of finding the maximum and minimum values?
• Graphical Representation: Plot the function $f(x) = 5^{x^2 - 4}$ on the interval $[3; 6]$ using a graphing calculator or software. This will provide a visual confirmation of our results. The graph will clearly show you the maximum and minimum points within the given range.

These exercises will not only deepen your understanding of the concepts but also enhance your problem-solving skills. So, keep practicing and experimenting, and you'll become even more comfortable with these types of mathematical challenges. Remember, math is a journey of continuous learning and exploration. Enjoy the process, and don't be afraid to tackle new problems! Keep practicing, guys; you're doing awesome! Keep exploring and have fun with it!