The gradient of a curve at the point $$\((x, y)\)$$ is given by $$\(\frac{\mathrm{d} y}{\mathrm{~d} x}=2(x+3)^{\frac{1}{2}}-x\)$$. The curve has a stationary point at $$\((a, 14)\)$$, where $$\(a\)$$ is a positive constant. Determine the nature of the stationary point. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m20_qp_12 Year：2020 Question No：10(b)
Answer:
\[
\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=(x+3)^{\frac{1}{2}}-1
\]
Sub their \(a \rightarrow \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{1}{3}-1=-\frac{2}{3}(\) or \(<0) \rightarrow \mathrm{MAX}\)
\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=(x+3)^{\frac{1}{2}}-1
\]
Sub their \(a \rightarrow \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{1}{3}-1=-\frac{2}{3}(\) or \(<0) \rightarrow \mathrm{MAX}\)
Knowledge points:
1.7.2 use the derivative of (for any rational ), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule
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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