Simplify & Analyze: F(x) = (x^(3/2) / X^(2/3) * √(2x^2)) * 2x^3

Hey guys! Let's dive into simplifying and analyzing a cool function today. We've got f(x) = (x^(3/2) / x^(2/3) * √(2x^2)) * 2x^3, and we need to break it down and see what it does within the range R = [1, 4]. This means we're looking at how the function behaves when x is between 1 and 4, inclusive. So, grab your thinking caps, and let's get started!

Simplifying the Function

Okay, first things first, let's simplify this beast of a function. Simplifying makes it way easier to understand and work with. We're going to use the basic rules of exponents and algebra to tidy things up. Remember, the goal here is to make the function as clean and manageable as possible. Here's the original function again:

f(x) = (x^(3/2) / x^(2/3) * √(2x^2)) * 2x^3

Step 1: Simplify the Square Root

Let's tackle the square root first. We have √(2x^2). We can rewrite this as:

√(2x^2) = √2 * √(x^2) = √2 * |x|

Now, since we're dealing with the range R = [1, 4], x is always positive in this range. This means we can drop the absolute value and simply write:

√2 * x

Step 2: Simplify the Fractional Exponents

Next up, we have x^(3/2) / x^(2/3). To simplify this, we use the rule of exponents which states that x^a / x^b = x^(a-b). So:

x^(3/2) / x^(2/3) = x^((3/2) - (2/3))

Now, we need to find a common denominator for the fractions 3/2 and 2/3. The common denominator is 6, so we rewrite the exponents:

(3/2) - (2/3) = (9/6) - (4/6) = 5/6

Thus:

x^(3/2) / x^(2/3) = x^(5/6)

Step 3: Combine the Simplified Terms

Now that we've simplified the square root and the fractional exponents, let's plug these back into our original function:

f(x) = (x^(5/6) * √2 * x) * 2x^3

Let's rearrange and combine the constants:

f(x) = 2√2 * x^(5/6) * x * x^3

Now, we combine the exponents of x. Remember, x is the same as x^1, so:

x^(5/6) * x * x^3 = x^(5/6) * x^1 * x^3 = x^((5/6) + 1 + 3)

Adding the exponents, we get:

(5/6) + 1 + 3 = (5/6) + (6/6) + (18/6) = 29/6

So, our function now looks like this:

f(x) = 2√2 * x^(29/6)

Simplified Function

Alright, guys! We've successfully simplified the function. The simplified form of f(x) is:

f(x) = 2√2 * x^(29/6)

This looks much cleaner and easier to handle than the original, right? Now that we've got this simplified form, we can move on to analyzing the function within the given range.

Analyzing the Function within the Range R = [1, 4]

Now that we've simplified the function to f(x) = 2√2 * x^(29/6), let's analyze how it behaves within the range R = [1, 4]. This means we want to understand what happens to the function's output (the y-values) as x varies between 1 and 4.

Step 1: Evaluate the Function at the Endpoints

First, let's evaluate the function at the endpoints of the range, which are x = 1 and x = 4. This will give us a good idea of the function's values at the boundaries.

Evaluating at x = 1

Plug in x = 1 into our simplified function:

f(1) = 2√2 * (1)^(29/6)

Since 1 raised to any power is still 1:

f(1) = 2√2 * 1 = 2√2

So, at x = 1, f(x) = 2√2. We can approximate this value: 2√2 ≈ 2 * 1.414 ≈ 2.828.

Evaluating at x = 4

Now, let's plug in x = 4:

f(4) = 2√2 * (4)^(29/6)

We can rewrite 4 as 2^2:

f(4) = 2√2 * (2²⁾(29/6)

Using the rule of exponents (a^b)c = a^(b*c):

f(4) = 2√2 * 2^(2 * (29/6)) = 2√2 * 2^(29/3)

Now, let's break this down a bit. We can rewrite 2^(29/3) as 2^(9 + 2/3) = 2^9 * 2^(2/3):

f(4) = 2√2 * 2^9 * 2^(2/3)

We know that 2^9 = 512. So:

f(4) = 2√2 * 512 * 2^(2/3)

We can rewrite 2^(2/3) as the cube root of 2 squared, which is the cube root of 4 (∛4). Approximating the values:

√2 ≈ 1.414 ∛4 ≈ 1.587

So:

f(4) ≈ 2 * 1.414 * 512 * 1.587 ≈ 2291.77

Thus, at x = 4, f(x) is approximately 2291.77. This gives us a sense of how much the function increases over the range.

Step 2: Analyze the Function's Behavior

We know f(x) = 2√2 * x^(29/6). Let's think about what this means:

• 2√2 is a constant, so it just scales the function. It doesn't affect the increasing or decreasing behavior.
• x^(29/6) is the crucial part. Since 29/6 is a positive exponent, the function will increase as x increases. This means as x moves from 1 to 4, f(x) will get larger.
• The exponent 29/6 is greater than 1 (it's approximately 4.83), which means the function increases at an accelerating rate. In other words, the function's graph will curve upwards.

Step 3: Check for Any Critical Points

Critical points are where the function's derivative is either zero or undefined. To find these, we need to take the derivative of f(x) and set it to zero. Let's find the derivative of f(x) = 2√2 * x^(29/6):

f'(x) = (29/6) * 2√2 * x^((29/6) - 1) f'(x) = (29/6) * 2√2 * x^(23/6)

Setting f'(x) to zero:

(29/6) * 2√2 * x^(23/6) = 0

Since (29/6) * 2√2 is a non-zero constant, the only way for the derivative to be zero is if x^(23/6) = 0, which means x = 0.

However, x = 0 is not in our range R = [1, 4]. Therefore, there are no critical points within our range. This means the function is either always increasing or always decreasing within the range. We already determined that the function is increasing, so we don't have to worry about any local maxima or minima within the range.

Step 4: Conclusion of Analysis

Alright, guys, let's wrap up our analysis. Within the range R = [1, 4], the function f(x) = 2√2 * x^(29/6):

• Is always increasing.
• Has a value of approximately 2.828 at x = 1.
• Has a value of approximately 2291.77 at x = 4.
• Has no critical points within the range.

This means the function starts at about 2.828 when x is 1, and it continuously increases to around 2291.77 when x is 4. The function increases at an accelerating rate, giving it a curved shape.

Final Thoughts

So, we've taken a pretty complex function, simplified it, and analyzed its behavior within a given range. This is a common task in calculus and helps us understand how functions behave in different situations. Remember, guys, simplifying first makes the analysis much easier! Keep practicing, and you'll become function-analyzing pros in no time!