How to Sketch the Curve of the Function f(x) 12x^5 - 45x^4 4x^3 5

Introduction to Curve Sketching

Curve sketching is an essential skill in mathematics, allowing us to visualize the behavior and characteristics of a function. This guide will walk you through the process of sketching the curve of the function f(x) 12x^5 - 45x^4 4x^3 5.

Understanding the Function

The function f(x) 12x^5 - 45x^4 4x^3 5 is a fifth-degree polynomial. As a polynomial of odd degree, it has specific end behaviors that are important to note:

As x rightarrow -infty, f(x) rightarrow -infty As x rightarrow infty, f(x) rightarrow infty

Key Components for Curve Sketching

1. Roots (x-intercepts): f(x) 0

Determining the x-intercepts involves solving the equation 12x^5 - 45x^4 4x^3 5 0. This is typically done using numerical methods or graphing tools. For sketching purposes, knowing the intervals where the roots lie is sufficient.

2. Local Min and Max: f'(x) 0

The first derivative, f'(x) 6^4 - 18^3 12x^2, helps identify the critical points where the function may have local minima or maxima. Solving f'(x) 0 gives:

6^4 - 18^3 12x^2 0 12x^2(5x^2 - 15x 1) 0 x 0, x 1, and x 2

These points are the critical points. Evaluating the function at these points gives:

f(0) 5 f(1) 12 f(2) -11

From these evaluations, we can determine:

A saddle point at x 0 A local maximum at x 1 A local minimum at x 2

3. Y-intercept: f(0)

The y-intercept is the value of the function when x 0, which is f(0) 5.

4. End Behavior: x rightarrow pminfty

The function ends at:

(-infty, -infty) (infty, infty)

Sketching the Curve

Let's sketch the curve step-by-step:

Step 1: Start with the End Behavior

Lightsly draw a diagonal line starting from the lower left corner in quadrant 3 and ending in the upper right corner in quadrant 1.

Step 2: Plot the Intercept

Plot the y-intercept at (0, 5).

Step 3: Identify the Critical Points

Plot the critical points (0, 5), (1, 12), and (2, -11). Flagging the positive and negative behavior in the intervals between these points.

Step 4: Find the Approximate Roots

Use Descartes' Rule of Signs to determine the number of positive and negative roots. Since there are 2 sign changes in the function, there are 2 positive roots. Testing values in the intervals will narrow down the roots:

f(-1) indicating a root between -1 and 0. Calculating f(1) > 0, f(2) , and f(3) > 0 pinpoint the roots to be in the interval (1, 3).

Step 5: Verify Using Technology (Optional)

To get a more precise view, use graphing software to plot the function and verify your sketch.

Advanced Techniques

Intersection Method for Zeroes

To find the approximate zeroes more accurately, plot the functions:

g(x) 12x^2 - 45x 40, and h(x) -5x^3 on a single graph. Identifying the intersections of these functions will give you the x-values where the zeroes of f(x) occur.

Slope Analysis

Evaluate the first derivative at the intersections to determine the slope and behavior (positive, negative, or zero).

Second Derivative for Concavity

Use the second derivative to identify inflection points and concavity.

Conclusion

By following these steps, you can effectively sketch the curve of the function f(x) 12x^5 - 45x^4 4x^3 5 and gain insights into its behavior and characteristics. This process can be refined further using numerical methods and graphing tools, making it a powerful skill for understanding complex functions.