Comprehensive Analysis Of F(x) = X⁴ − 4x³ Critical Points, Intervals, And Extrema

Hey guys! Today, we're diving deep into the fascinating world of calculus to explore the function f(x) = x⁴ − 4x³. We'll be uncovering its critical points, mapping out where it's increasing and decreasing, and pinpointing those all-important maximum and minimum values. So, buckle up and let's get started on this mathematical adventure!

i) Finding the Critical Points of f(x) = x⁴ − 4x³

Let's start by defining critical points. In the realm of calculus, critical points are the x-values where the derivative of a function, f'(x), is either equal to zero or undefined. These points are crucial because they often mark the spots where a function changes its behavior – think peaks, valleys, or even flat stretches. In simpler terms, critical points are the potential turning points of our function's graph.

To find these critical points, our first step is to calculate the derivative of f(x). Remember the power rule? It's our trusty tool here. Applying it to f(x) = x⁴ − 4x³, we get:

f'(x) = 4x³ − 12x²

Now, we need to find where this derivative equals zero. This means solving the equation:

4x³ − 12x² = 0

Let's factor out a common term, 4x²:

4x²(x − 3) = 0

This gives us two possible solutions:

• 4x² = 0 => x = 0
• x − 3 = 0 => x = 3

So, there you have it! Our critical points are x = 0 and x = 3. These are the x-values we need to investigate further to understand the function's behavior. To finalize our analysis, let's find the corresponding y-values by plugging these x-values back into our original function, f(x) = x⁴ − 4x³:

• f(0) = (0)⁴ − 4(0)³ = 0
• f(3) = (3)⁴ − 4(3)³ = 81 − 108 = -27

Therefore, the critical points, expressed as coordinates, are (0, 0) and (3, -27). These points are significant landmarks on the graph of f(x), indicating potential local maxima, local minima, or points of inflection. Understanding these critical points is pivotal in sketching the graph of the function and comprehending its overall behavior. We're building a solid foundation for our analysis, guys!

ii) Unveiling the Sign of f'(x) and Mapping Increasing and Decreasing Intervals

Now, let's use the first derivative test to figure out where our function is increasing and decreasing. This test is a powerful way to understand a function's behavior based on the sign of its derivative. Remember, a positive f'(x) means the function is climbing uphill (increasing), while a negative f'(x) means it's sliding downhill (decreasing). To do this, we'll create a sign chart using our critical points as dividers. The critical points we found earlier, x = 0 and x = 3, divide the number line into three intervals:

1. (−∞, 0)
2. (0, 3)
3. (3, ∞)

We'll pick a test value within each interval and plug it into our derivative, f'(x) = 4x³ − 12x², to see if it's positive or negative:

1. Interval (−∞, 0): Let's choose x = -1 f'(-1) = 4(-1)³ − 12(-1)² = -4 − 12 = -16 (Negative) This indicates that the function f(x) is decreasing in the interval (−∞, 0).
2. Interval (0, 3): Let's choose x = 1 f'(1) = 4(1)³ − 12(1)² = 4 − 12 = -8 (Negative) This means f(x) is also decreasing in the interval (0, 3).
3. Interval (3, ∞): Let's choose x = 4 f'(4) = 4(4)³ − 12(4)² = 256 − 192 = 64 (Positive) This signifies that f(x) is increasing in the interval (3, ∞).

Now, let's summarize our findings in a table, which clearly shows the intervals, test values, sign of f'(x), and the behavior of f(x):

From this sign analysis, we can see that:

• f(x) is decreasing on the intervals (−∞, 0) and (0, 3).
• f(x) is increasing on the interval (3, ∞).

This information gives us a clear picture of how the function behaves across its domain. We know where it's heading downhill and where it's climbing uphill. Understanding the sign of the derivative provides essential insights into the function's graph, allowing us to sketch it accurately and predict its behavior. This is super helpful, right?

iii) Spotting Maximums and Minimums: Unveiling the Extremes of f(x)

Alright, let's get to the exciting part – finding the maximum and minimum points of our function! We'll use the information we gathered from the sign of the derivative in the previous section to identify these extreme values. Remember, we're looking for points where the function switches from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum). The critical points we found earlier are the key locations to investigate for these changes.

Looking back at our sign chart, we can see the following behavior around our critical points:

• At x = 0: The function is decreasing before x = 0 and continues to decrease after x = 0. This means there's no change in direction here. It's neither a local maximum nor a local minimum. It’s actually a point of inflection, where the concavity of the graph changes.
• At x = 3: The function is decreasing before x = 3 and then starts increasing after x = 3. This indicates a change from decreasing to increasing, which means we have a local minimum at x = 3.

So, we've identified a local minimum at x = 3. To find the y-coordinate of this minimum point, we plug x = 3 back into our original function, f(x) = x⁴ − 4x³:

f(3) = (3)⁴ − 4(3)³ = 81 − 108 = -27

Therefore, we have a local minimum at the point (3, -27).

Now, let's discuss the existence of global (or absolute) maximums and minimums. A global maximum is the highest point on the entire graph of the function, while a global minimum is the lowest point. Since our function is a polynomial of degree 4 with a positive leading coefficient, we know that as x approaches positive or negative infinity, f(x) also approaches positive infinity. This means there is no global maximum for this function. The function will keep increasing without bound as we move away from the critical points.

However, we do have a local minimum at (3, -27), and since the function increases towards infinity on both ends, this local minimum is also the global minimum for the function. There is no other point on the graph of f(x) that is lower than -27.

In summary:

• Local Minimum: (3, -27)
• Global Minimum: (3, -27)
• Local Maximum: None
• Global Maximum: None

And there you have it, guys! We've successfully navigated the world of f(x) = x⁴ − 4x³, uncovering its critical points, mapping its increasing and decreasing intervals, and pinpointing its minimum value. By using the power of calculus, we've gained a deep understanding of this function's behavior. Keep exploring, keep questioning, and keep having fun with math!