1.16: Logarithmic curve: derivative and stationary point

0606/11/O/N/17 — Question 7 · 9 marks

A curve has equation

y = \ln\left(\dfrac{2x + 1}{2x - 1}\right) + 4x

for

x > \dfrac{1}{2}

(i)

Write

\ln\left(\dfrac{2x + 1}{2x - 1}\right)

as the difference of two logarithms.

[1]

(ii)

Using your answer to part (i) show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{ax^{2} + b}{4x^{2} - 1}

, where

a

and

b

are integers.

[4]

(iii)

Hence find the

x

-coordinate of the stationary point on the curve.

[2]

(iv)

Determine the nature of this stationary point.

[2]