Verify that this is true for \(f(x) = {x^3} + 1\) at x = 2.

Given that \(g(x) = x{{\text{e}}^{{x^2}}}\), show that \(g'(x) > 0\) for all values of x.

Using the result given at the start of the question, find the value of the gradient function of \(y = {g^{ - 1}}(x)\) at x = 2.

(i)     With f and g as defined in parts (a) and (b), solve \(g \circ f(x) = 2\).

(ii)     Let \(h(x) = {(g \circ f)^{ - 1}}(x)\). Find \(h'(2)\).

\(f(2) = 9\)     (A1)

\({f^{ - 1}}(x) = {(x - 1)^{\frac{1}{3}}}\)     A1

\(({f^{ - 1}})'(x) = \frac{1}{3}{(x - 1)^{ - \frac{2}{3}}}\)     (M1)

\(({f^{ - 1}})'(9) = \frac{1}{{12}}\)     A1

\(f'(x) = 3{x^2}\)     (M1)

\(\frac{1}{{f'(2)}} = \frac{1}{{3 \times 4}} = \frac{1}{{12}}\)     A1

Note: The last M1 and A1 are independent of previous marks.

[6 marks]

\(g'(x) = {{\text{e}}^{{x^2}}} + 2{x^2}{{\text{e}}^{{x^2}}}\)     M1A1

\(g'(x) > 0\) as each part is positive     R1

[3 marks]

to find the x-coordinate on \(y = g(x)\) solve

\(2 = x{{\text{e}}^{{x^2}}}\)     (M1)

\(x = 0.89605022078 \ldots \)     (A1)

gradient \( = ({g^{ - 1}})'(2) = \frac{1}{{g'(0.896 \ldots )}}\)     (M1)

\( = \frac{1}{{{{\text{e}}^{{{(0.896 \ldots )}^2}}}\left( {1 + 2 \times {{(0.896 \ldots )}^2}} \right)}} = 0.172\) to 3sf     A1

(using the \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) function on gdc \(g'(0.896 \ldots ) = 5.7716028 \ldots \)

\(\frac{1}{{g'(0.896 \ldots )}} = 0.173\)

[4 marks]

(i)     \(({x^3} + 1){{\text{e}}^{{{({x^3} + 1)}^2}}} = 2\)     A1

\(x = - 0.470191 \ldots \)     A1

(ii)     METHOD 1

\((g \circ f)'(x) = 3{x^2}{{\text{e}}^{{{({x^3} + 1)}^2}}}\left( {2{{({x^3} + 1)}^2} + 1} \right)\)     (M1)(A1)

\((g \circ f)'( - 0.470191 \ldots ) = 3.85755 \ldots \)     (A1)

\(h'(2) = \frac{1}{{3.85755 \ldots }} = 0.259{\text{ }}(232 \ldots )\)     A1

Note: The solution can be found without the student obtaining the explicit form of the composite function.

METHOD 2

\(h(x) = ({f^{ - 1}} \circ {g^{ - 1}})(x)\)     A1

\(h'(x) = ({f^{ - 1}})'\left( {{g^{ - 1}}(x)} \right) \times ({g^{ - 1}})'(x)\)     M1

\( = \frac{1}{3}{\left( {{g^{ - 1}}(x) - 1} \right)^{ - \frac{2}{3}}} \times ({g^{ - 1}})'(x)\)     M1

\(h'(2) = \frac{1}{3}{\left( {{g^{ - 1}}(2) - 1} \right)^{ - \frac{2}{3}}} \times ({g^{ - 1}})'(2)\)

\( = \frac{1}{3}{(0.89605 \ldots  - 1)^{ - \frac{2}{3}}} \times 0.171933 \ldots \)

\( = 0.259{\text{ }}(232 \ldots )\)     A1     N4

[6 marks]

There were many good attempts at parts (a) and (b), although in (b) many were unable to give a thorough justification.

There were many good attempts at parts (a) and (b), although in (b) many were unable to give a thorough justification.

Few good solutions to parts (c) and (d)(ii) were seen although many were able to answer (d)(i) correctly.

Few good solutions to parts (c) and (d)(ii) were seen although many were able to answer (d)(i) correctly.