11 The equation of a curve is \(y = k x ^ { \frac { 1 } { 2 } } - 4 x ^ { 2 } + 2\), where \(k\) is a constant.
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(k\).
- It is given that \(k = 2\).
Find the coordinates of the stationary point and determine its nature.
\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-16_2717_38_110_2010} - Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6 .
Find this value of \(k\).
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If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-19_2717_35_106_20}