Find the x coordinate of the point of maximum curvature (call it x0 ) on the curve y=3e^5x and find the maximum curvature, κ(x0)

Answer 1

made x and K more accurate

K=75e^5x/(1+225e^10x)^(3/2) from formula for K

if you graph K it has a max at about -.7=x

the value of K from the graph at x=-.7 is K=1.7

approximately.

calculating dK/dx and setting to zero is a bit complicated.

Answer 2

if by "curvature" you mean slope, there is no maximum, it approaches infinity, as x approaches infinity. The slope approaches + infinity

as x approaches negative infinity, the slope approaches zero, the x axis is a horizontal asymptote

curvature = 2nd derivative of y with respect to x, all divided by one minus same, which gives most people a migraine headache or close to it