find a and b constants

Question

find the constants a and b in the function f(x) =(ax^b)/ln(x) such that f(1/2) = 1 and the function has a local minimum at x=1/2.

a=

b=

Answer 1

x = e^1/b means ln(x) = 1/b, so

f ´(½) = 0 means ln(½) = –ln(2) = 1/b

So b = –1/ln(2)

f(½) = 1 means ae^–bln(2) = –ln(2)

Since b = –1/ln(2), this means –bln(2) = 1

So ae^–bln(2) = ae = –ln(2)

and

a = –ln(2)/e

Desmos plot: desmos.com/calculator/vfzyypzswi

Answer 2

desmos.com/calculator/er9zq3uec4