The diagram shows part of the curve with equation $$\(y=x^{\frac{1}{2}}+k^{2} x^{-\frac{1}{2}}\)$$, where $$\(k\)$$ is a positive constant. Find the coordinates of the minimum point of the curve, giving your answer in terms of $$\(k\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_13 Year：2021 Question No：11(a)
Answer:
\(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{2} x^{-1 / 2}-\frac{1}{2} k^{2} x^{-3 / 2}\)
\(\frac{1}{2} x^{-1 / 2}-\frac{1}{2} k^{2} x^{-3 / 2}=0 \quad\) leading to \(\frac{1}{2} x^{-1 / 2}=\frac{1}{2} k^{2} x^{-3 / 2}\)
\(\left(k^{2}, 2 k\right)\)
\(\frac{1}{2} x^{-1 / 2}-\frac{1}{2} k^{2} x^{-3 / 2}=0 \quad\) leading to \(\frac{1}{2} x^{-1 / 2}=\frac{1}{2} k^{2} x^{-3 / 2}\)
\(\left(k^{2}, 2 k\right)\)
Knowledge points:
1.1.5 recognise and solve equations in which are quadratic in some function of
1.7.4 locate stationary points and determine their nature, and use information about stationary points in sketching graphs. (Including use of the second derivative for identifying maxima and minima; alternatives may be used in questions where no method is specified.) (Knowledge of points of inflexion is not included.)
Solution:
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