\( \lim _ { x \rightarrow 0 } 2 x \sin ( \frac { 1 } { x } ) - \cos ( \frac { 1 } { x } ) \)

กำหนดฟังก์ชัน \( \mathrm{f}(\mathrm{x})=2 \mathrm{x}^{3}-9 \mathrm{x}^{2}+12 \mathrm{x}-2 \) จุดต่ำสุดสัมพัทธ์ เท่ากับคือข้อใด \( \begin{array}{ll}\text { 1) }\left(\frac{3}{2}, \frac{5}{2}\right) & \text { 2) }\left(\frac{1}{2}, \frac{3}{2}\right) \\ \text { 3) }(1,3) & \text { 4) }(2,2)\end{array} \)

We define the function \( f(x)=\frac{\sin (3 x)}{x} \), where \( \sin (\theta) \) applies to \( \theta \) given in radians. (a) Which of the following best describes the value of \( f(0) \) ? (No answer given)

\( y ^ { \prime \prime } - 2 y ^ { \prime } + y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = \)

a) Consider the equation \[ \frac{2 t y}{t^{2}+1}-2 t-\left(2-\ln \left(t^{2}+1\right)\right) y^{\prime}=0 \] i. Determine whether the differential equation is exact. ii. Find every solution of the equation above. ( 6 Marks) b) Find the general solution to the following Bernoulli Equation \( y^{\prime}+x y=x y^{3} \). ( 6 Marks) Find the solution to the following homogeneous differential equation. ( 5 Marks)

\( f(x)=\frac{x^{2}-4 x+7}{1-x} \) demantre que \( f^{\prime}(x)=\frac{4-(x-1)^{2}}{(x-1)^{2}} \) pan \( x \in \mathbb{R}-\{1\} \).