At what point does the curve have maximum curvature? y = 7 Inx

Solution:

Curvature K  =    |f’’(x)| / [1 + f’(x)²]^3/2

Given f(x) = 7 ln x

f’(x) = 7/x

f’’(x) = -1/x²

Therefore, curvature K = ( 1/x²) / [1+ (7/x)²]^3/2

=  1/x³[1+ 49/x²]^3/2

=   x/[x² + 49]^3/2

To find the maximum curvature, let us differentiate the above equation.

dK/dx  = ( [49 + x²]^3/2 dx/dx  - (x)(3/2)(2x)[49 +  x²]^1/2 ) / [49 + x²]³

=  1/[49 + x²]^3/2 - 3x²/[49 + x²]^5/2

= (49 + x²) -  3x²

= [49 + x²]^5/2

= 49 - 2x²

= [49 + x²]^5/2

For maximum curvature, dK/dx has to be zero. The above expression will be zero: 

When x = 7/√2 or 4.949

On either side of x = 7/√2 

K ≷ 0

Therefore the maximum curvature will be at x = 7/√2  or 4.949

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At what point does the curve have maximum curvature? y = 7 Inx

Summary:

The point at which  the curve y = lnx will have the maximum curvature will be at x = 7/√2. The y value will be  ln(7/√2) = 1.6  at that point.