Show that the \(x\)-coordinate of the minimum point on the curve \(y = f(x)\) satisfies the equation \(\tan x = 2x\).

Determine the values of \(x\) for which \(f(x)\) is a decreasing function.

Sketch the graph of \(y = f(x)\) showing clearly the minimum point and any asymptotic behaviour.

Find the coordinates of the point on the graph of \(f\) where the normal to the graph is parallel to the line \(y =  - x\).

This region is now rotated through \(2\pi \) radians about the \(x\)-axis. Find the volume of revolution.

attempt to use quotient rule or product rule     M1

\(f’(x) = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}{\text{ }}\left( { = \frac{1}{{2\sqrt x \sin x}} - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}} \right)\)     A1A1

Note:     Award A1 for \(\frac{1}{{2\sqrt x \sin x}}\) or equivalent and A1 for \( - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}\) or equivalent.

setting \(f’(x) = 0\)     M1

\(\frac{{\sin x}}{{2\sqrt x }} - \sqrt x \cos x = 0\)

\(\frac{{\sin x}}{{2\sqrt x }} = \sqrt x \cos x\) or equivalent     A1

\(\tan x = 2x\)     AG

[5 marks]

\(x = 1.17\)

\(0 < x \leqslant 1.17\)     A1A1

Note:     Award A1 for \(0 < x\) and A1 for \(x \leqslant 1.17\). Accept \(x < 1.17\).

[2 marks]

concave up curve over correct domain with one minimum point above the \(x\)-axis.     A1

approaches \(x = 0\) asymptotically     A1

approaches \(x = \pi \) asymptotically     A1

Note:     For the final A1 an asymptote must be seen, and \(\pi \) must be seen on the \(x\)-axis or in an equation.

[3 marks]

\(f’(x){\text{ }}\left( { = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}} \right) = 1\)     (A1)

attempt to solve for \(x\)     (M1)

\(x = 1.96\)     A1

\(y = f(1.96 \ldots )\)

\( = 1.51\)     A1

[4 marks]

\(V = \pi \int_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{{x{\text{d}}x}}{{{{\sin }^2}x}}} \)     (M1)(A1)

Note:     M1 is for an integral of the correct squared function (with or without limits and/or \(\pi \)).

\( = 2.68{\text{ }}( = 0.852\pi )\)     A1

[3 marks]