Final answer:
The curve y = 3 ln x has maximum curvature immediately to the right of x = 0. This occurs because the second derivative, which dictates curvature, has its greatest value in magnitude as x approaches zero from the right. Due to the undefined nature of the natural logarithm at x = 0, the point of maximum curvature cannot be exactly reached.
Step-by-step explanation:
To determine at what point the curve y = 3 ln x has maximum curvature, we first need to find its second derivative. The curvature of a curve at a given point is determined by the second derivative of the function representing the curve.
Let's start by differentiating y = 3 ln x. The first derivative, using the chain rule, is:
y' = 3/x
Now, we'll differentiate again to find the second derivative
y'' = -3/x^2
Curvature is related to the absolute value of the second derivative. Since -3/x^2 is always negative for positive x and decreases in absolute value as x increases, the maximum curvature occurs as x approaches zero from the right.
However, because the natural logarithm is undefined for x <= 0, we cannot use x = 0. Therefore, the curve y = 3 ln x has maximum curvature immediately to the right of x = 0, or as close to zero as possible without actually being zero.
In the real world, we cannot reach an exact point of maximum curvature on this function since the logarithm of zero is undefined and the function's domain is restricted to positive real numbers. The concept of approaching a point is an important aspect of calculus when examining limits and behavior near points where the function itself might not be well-defined.