If 2 curves #y=kx^n# and #y=lnx# have the same gradient exactly at x=a, find the relationship between a, k and n? Also, if these 2 curves also intersect at the point x=a, express k in terms of n?

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Answer 1 · 2016-10-12T16:38:51

#a = e^(1/n) and k =1 / (n e) #

The two intersect at x = a..

So, the common #y at x = a =ka^n=ln a#

At this common point, y' is said to be the same.. So,

#kna^(n-1)=1/a#. And so, from the two relations governing k, n and a,

#a^n=1/(kn)=ln a/k#. It follows that

#n ln a = 1. to a = e(1/n) and k = 1/e#

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