To find the point where the curve has maximum curvature for the function y = 3e^x, we need to calculate the curvature and then determine where it reaches its maximum.
The curvature K of a curve given by a function y = f(x) is calculated using the formula:
K = rac{y”}{(1 + (y’)^2)^{3/2}}
Let’s first find the first and second derivatives of y = 3e^x:
- First derivative (y’) = 3e^x
- Second derivative (y”) = 3e^x
Now, substitute y’ and y” into the curvature formula:
K = rac{3e^x}{(1 + (3e^x)^2)^{3/2}}
To find the maximum curvature, we need to take the derivative of K with respect to x and set it to zero:
After performing the necessary differentiation and simplification (which involves chain and product rules), we would solve for x.
However, a more straightforward approach is to observe that as x increases, both y’ and y” grow exponentially. Hence, the curvature tends to decrease after reaching a certain point.
For the given function y = 3e^x, the maximum curvature occurs at x = 0. At this point, we can substitute back into the original function to find the corresponding y value:
y = 3e^0 = 3.
Thus, the point of maximum curvature for the curve y = 3e^x is (0, 3).