Understanding the Inflection Points of the Function f(x) x3 ln x

Inflection points are crucial in understanding the behavior of functions and the curvature of their graphs. In this article, we will delve into the inflection point of the function f(x) x3 ln x, demonstrating the steps to find and interpret these points. This knowledge is valuable for analyzing the function's concavity and optimizing its behavior in various applications.

Introduction to Inflection Points

An inflection point on a curve is a point where the curve changes its concavity. In other words, it is a point where the second derivative of the function changes its sign. This can be visualized as the point where a curve transitions from being concave up (U-shaped) to concave down (reverse U-shaped), or vice versa.

Deriving the Function

Let's start with the given function:

[y x^3 ln(x)]

To find the inflection points, we need to derive the second derivative of the function. Let's break down the process step-by-step.

First Derivative

Using the product rule, we have:

[f'(x) 3x^2 ln(x) x^3 cdot frac{1}{x}]

Simplifying the expression, we get:

[f'(x) 3x^2 ln(x) x^2]

Second Derivative

Now, we need to find the second derivative:

[f''(x) 6x ln(x) 3x^2 cdot frac{1}{x} 2x]

Simplifying it, we have:

[f''(x) 6x ln(x) 3x 2x]

Combining like terms, we get:

[f''(x) 6x ln(x) 5x]

Further Simplification

We can further simplify the expression:

[f''(x) x(6 ln(x) 5)]

Setting the second derivative to zero to find the inflection points, we get:

[x(6 ln(x) 5) 0]

Solving for x, we have two cases:

[x 0], which is not in the domain of the original function since (ln(0)) is undefined. [6 ln(x) 5 0]

Finding the Inflection Point

Solving the equation:

[6 ln(x) 5 0]

[6 ln(x) -5]

[(ln(x) -frac{5}{6})]

[x e^{-frac{5}{6}}]

Substituting (x e^{-frac{5}{6}}) back into the original function to find the y-coordinate:

[y (e^{-frac{5}{6}})^3 ln(e^{-frac{5}{6}})]

[y e^{-frac{5}{2}} left(-frac{5}{6}right)]

[y -frac{5}{6} e^{-frac{5}{2}}]

Therefore, the inflection point of the function is at:

[left(e^{-frac{5}{6}}, -frac{5}{6} e^{-frac{5}{2}}right)]

Conclusion

The inflection point of the function f(x) x3 ln x is at (x e^{-frac{5}{6}}), and the corresponding y-coordinate is (-frac{5}{6} e^{-frac{5}{2}}). This point marks a significant change in the curvature of the function, indicating where the concavity switches from one type to another.

Related Keywords

inflection point second derivative concavity mathematical function