The curvature of a curve at any given point measures how sharply it bends. For the function y = x ln(x), we can analyze its curvature to find the point where it reaches a maximum value and how this curvature behaves as x approaches infinity.
To find the maximum curvature, we need to calculate the curvature (K) of the function, which can be expressed as:
K = rac{y''}{(1 + (y')^2)^{3/2}}Here, y’ is the first derivative of the function and y” is the second derivative. Let’s calculate these derivatives:
1. The first derivative y’ of y = x ln(x) is:
y' = ln(x) + 12. The second derivative y” is:
y'' = rac{1}{x}Now, substituting these values into the curvature formula:
K = rac{ rac{1}{x}}{(1 + (ln(x) + 1)^2)^{3/2}}To find the maximum curvature, we need to analyze the behavior of K with respect to x. We will differentiate K with respect to x and set the result to zero. However, instead of going through the complicated differentiation process, we can assess the limits directly.
As x approaches infinity, the term ln(x) grows, leading to an increase in both y’ and (y’)^2. This implies that:
1 + (ln(x) + 1)^2 o ext{infinity}This will cause the denominator of the curvature K to increase significantly. Meanwhile, the second derivative y” approaches zero since:
y'' = rac{1}{x} o 0 ext{ as } x o ext{infinity}Thus, K approaches:
K o 0 ext{ as } x o ext{infinity}This means the curvature of the curve y = x ln(x) decreases and approaches zero, indicating that the curve becomes less sharp and more linear as x tends to infinity.
In summary, the curve of y = x ln(x) achieves its maximum curvature at a certain finite point, which we can find by evaluating the curvature function and determining where its derivative is zero. As x approaches infinity, the curvature tends towards zero, suggesting that the curve flattens out as it extends outward.