Question

Show that the function given by has maximum at x = e .

Answer 1

The function is given as, f(x)= logx x (1)

Differentiate the equation,

f ′ (x)= x⋅ 1 x −logx⋅1 x 2 = 1−logx x 2

Again differentiate the above equation,

f ′ ′ (x)= x 2 ⋅( − 1 x )−( 1−logx )⋅2x x 4 = −x−2x+2xlogx x 4 = x( 2logx−3 ) x 4 = 2logx−3 x 3 (2)

For maximum value,

f ′ ( x )=0 1−logx x 2 =0 logx=1 x=e

From equation (2)

f ′ ′ (x)= 2loge−3 e 3 = −1 e 3 <0

Thus, x=e is a point of local maxima and maximum value of function is at x=e.