The equation of a curve is $$\(y=2 x+1+\frac{1}{2 x+1}\)$$ for $$\(x>-\frac{1}{2}\)$$. Find the coordinates of the stationary point and determine the nature of the stationary point. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w20_qp_13 Year：2020 Question No：8(b)
Answer:
Set their \(\frac{\mathrm{d} y}{\mathrm{~d} x}=0\) and attempt solution
\((2 x+1)^{2}=1 \rightarrow 2 x+1=(\pm) 1\) or \(4 x^{2}+4 x=0 \rightarrow(4) x(x+1)=0\)
\(x=0\)
\((0,2)\)
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}>0\) from a solution \(x>-\frac{1}{2}\) hence minimum
\((2 x+1)^{2}=1 \rightarrow 2 x+1=(\pm) 1\) or \(4 x^{2}+4 x=0 \rightarrow(4) x(x+1)=0\)
\(x=0\)
\((0,2)\)
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}>0\) from a solution \(x>-\frac{1}{2}\) hence minimum
Knowledge points:
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
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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