1.14: Exponential inverse and sketch of quadratic

0606/13/M/J/19 — Question 8 · 10 marks

The functions

f

and

g

are defined for real values of

x

f : x \mapsto \mathrm{e}^{3x}

and

g : x \mapsto 2x^{2} + 1

for

x \geqslant 0

(i)

Write down the range of

g

[1]

(ii)

Show that

f^{-1}g(\sqrt{62}) = \ln 5

[3]

(iii)

Solve

f'(x) = 6g''(x)

, giving your answer in the form

\ln a

, where

a

is an integer.

[3]

(iv)

On the axes below, sketch the graph of

y = g

and the graph of

y = g^{-1}

, showing the points where the graphs meet the coordinate axes.

[3]